faculty of economics · 2020. 5. 20. · cambridge-inet working paper series no: 2017/02 cambridge...
TRANSCRIPT
Cambridge-INET Working Paper Series No: 2017/02
Cambridge Working Papers in Economics: 1704
THE STRATEGY OF CONQUEST
Marcin Dziubiński Sanjeev Goyal David E. N. Minarsch (University of Warsaw) (University of Cambridge) (Fetch.AI.)
(revised 11 March 2020) This paper develops a theoretical framework for the study of war and conquest. The analysis highlights the role of three factors - the technology of war, resources, costs of war, alliances, and contiguity network - in shaping the dynamics of appropriation and the formation of empires. Keywords: Empire, conflict, contiguity network, resources.
Cambridge Working Papers in Economics
Faculty of Economics
The Strategy of Conquest
Marcin Dziubinski∗ Sanjeev Goyal† David E. N. Minarsch‡
March 11, 2020
Abstract
This paper develops a theoretical framework for the study of war and conquest. The
analysis highlights the role of three factors – the technology of war, resources, costs of
war, alliances, and contiguity network – in shaping the dynamics of appropriation and
the formation of empires.
Keywords Empire, conflict, contiguity network, resources.
∗Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Email:
[email protected]†Faculty of Economics and Christ’s College, University of Cambridge. Email: [email protected]‡Fetch.AI. Email: [email protected].
We are grateful to Sebastian Cortes and Gustavo Paez for expert research assistance and to Leonie Bau-
mann, Sihua Ding, Matt Elliott, Edoardo Gallo, Sng Tuan Hwee, Jorg Kalbfuss, and Bryony Reich for
detailed comments on earlier drafts of the paper. The authors are indebted to Daron Acemoglu, Francis
Bloch, Gabriel Carroll, Vince Crawford, Bhaskar Dutta, Joan Esteban, James Fearon, Mark Harrison, John
Geanakoplos, Kai Konrad, Dan Kovenock, Dunia Lopez-Pintado, Kai Konrad, Meg Meyer, Herve Moulin,
Brian Rogers, Sheilagh Ogilvie, Wojciech Olszewski, Robert Powell, Debraj Ray, Stefano Recchia, Phil Reny,
Stergios Skaperdas, Bruno Strulovici, Muhamet Yildiz, Ayse Zarakol, and participants at a number of seminars
for suggestions and encouragement. Dziubinski acknowledges support from Polish National Science Centre
through Grant 2014/13/B/ST6/01807. Goyal acknowledges support from a Keynes Fellowship. Goyal and
Minarsch acknowledge support from the Cambridge-INET Institute.
1 Introduction
The history of the world .... is an imperial history, the history of empires. Empires
were systems of influence or rule where ethnic, cultural or ecological boundaries
were overlapped or ignored. Their ubiquitous presence arose from the fact that
.... the endowments needed to build strong states were very unequally distributed.
Against the cultural attraction, or physical force, of an imperial state, resistance
was hard, unless reinforced by geographical remoteness or unusual cohesion. (Dar-
win [2007]; page 491)
A recurring theme in history is that the presence of small kingdoms is accompanied by
bloody conflict; rulers fight each other incessantly, small parcels of land are exchanged, trea-
sures are plundered, and capture of human beings is common. However, once a ruler acquires
a large advantage relative to his neighbours, he then quickly goes on to take them over, one
after the other, and to create an empire.1 This record of war and conquest leads us to ask:
What are the circumstances under which rulers will choose to fight? What is the optimal
timing of attack, now or later? When will the resource advantage of a ruler translate into
domination over neighbours? What are the limits to the size of the empire? The goal of this
paper is to develop a theoretical framework to address these questions.
We consider a set of ‘kingdoms’. Every kingdom is endowed with resources and controlled
by a ruler. Rulers desire to expand territory and acquire more resources. The ruler can wage
a war on neighboring kingdoms. The winner of a war takes control of the loser’s resources
and his kingdom; the loser is eliminated. The probability of winning a war depends on the
resources of the combatants and on the technology of war that is defined by a contest success
function.2 As the winning ruler expands his domain, he may be able to access and attack new
kingdoms. The neighborhood structure between kingdoms is reflected in a contiguity network.
We model the interaction between rulers as a dynamic game and study its (Markov Perfect)
equilibria.
We start by establishing that there exists a pure strategy Markov Perfect equilibrium
1Classical studies on the formation of empire include Polybius [2010], Tacitus [2009] and Khaldun [1989].Starting with Gibbon [1776], there is a long tradition of modern work on empires, see e.g., Braudel [1995],Darwin [2007], Elliott [2006], Lewis [2010], Morris and Scheidel [2009], and Thapar [1997, 2002]. Mathematicalmodels of the evolution of empire include Krainin and Wiseman [2016], Levine and Modica [2013], and Turchin[2007].
2Classical writers on war and more recent research both point to the decisive role of the army size andfinancial resources in securing victory, see e.g., Lewis [2010], Tzu [2008], Clausewitz [1993] and Howard [2009].
1
and the equilibrium payoffs are unique. This sets the stage for a study of how the main
parameters—resources, the contiguity network, and the contest function—affect the dynamics
of war and peace.
Consider two rulers A and B, with resources xA and xB, and suppose xA > xB. When
they fight, the expected payoff of A is given by (xA + xB)p(xA, xB), where p(xA, xB) is the
contest success function that defines the probability of winning for ruler A. The contest success
function is said to be rich rewarding if fighting is profitable for A (and unprofitable for B),
i.e., (xA + xB)p(xA, xB) > xA. The technology is said to be poor rewarding, otherwise. The
technology shapes the optimal timing and the target of attack. When the technology is rich
rewarding, no-waiting is optimal: attacking the two rivals in sequence is preferable to attacking
the merged kingdom. In the poor rewarding setting, waiting is optimal: attacking the larger
kingdom formed after two rivals have fought is best. Moreover, with a rich (poor) rewarding
technology it is optimal for a ruler to attack opponents in increasing (decreasing) order of
resources. Equipped with these results, we turn to the study of equilibrium dynamics.
Theorem 1 shows that, with a rich rewarding technology, in any configuration with three
or more kingdoms, all rulers, even the poorest ones, find it optimal to attack a neighbour as
soon as possible. Thus, we are in a world with incessant warfare, the violence only stops when
all opposition is eliminated. When the network is connected, all opposition is eliminated only
with the hegemony of a single ruler.3 The arguments underlying this result are fairly general.
We start by defining a strong ruler: this is a ruler who has a ‘full attacking sequence’ (involving
all other opponents), such that at each point he is stronger than the opponent. Clearly, at any
point in time, the richest ruler is a strong ruler. It follows from the rich rewarding property
that, if everyone else is peaceful, then such a strong ruler has a strict incentive to fight every
other ruler. Next consider the case when other rulers may also wish to attack: does the strong
ruler still have an incentive to implement a fully attacking sequence? Given the no-waiting
property identified above, it then follows that the strong ruler has a dominant strategy: a full
attacking sequence. So, there is always at least one ruler who wishes to fight to the finish.
Anticipating this, and given the no-waiting property, every ruler, no matter how poor, has an
incentive to fight a neighbour. Thus in a connected network, in equilibrium, eventually there
will be only one ruler left. Remarkably, this result does not depend on the topology of the
network, as long as it is connected. Even very sparsely connected networks cannot prevent
conflict escalation.4
3A network is connected if there is a path between any two kingdoms.4Earlier works of Krainin and Wiseman [2016] and Levine and Modica [2013] obtain the hegemony result
2
Turning to the role of resources and networks in shaping the prospects of individual rulers,
firstly the network, via the neighborhood relation, determines access to weaker kingdoms and
threat from stronger ones. A ruler with relatively low resources my be strong because the
network enables him to accumulate resources by fighting weaker opponents. On the other
hand, a ruler with relatively high resources may be surrounded by stronger opponents and
have very limited opportunities to become a hegemon. This is the basic level on which
the distribution of resources, the topology of the network, and the position in the network
determines the probability of becoming a hegemon. For ease of exposition, consider the
well known Tullock Contest Function: the probability of ruler A winning is p(xA, xB) =
xγA/(xγA + xγB), for some γ ∈ R+. It can be shown that the function is rich rewarding if γ > 1
and poor rewarding if γ < 1 (and rulers are indifferent between war and peace if γ = 1).
When γ is sufficiently large, the probability of a weak ruler becoming a hegemon becomes
negligible and the key factor affecting the probability of becoming a hegemon is whether the
rulers is strong or weak. Within the set of strong rulers, those who have ‘exclusive’ access to
weak kingdoms, have a significantly greater probability of becoming the hegemon (relative to
their strong rivals). We show that the dynamics of appropriation have powerful redistribution
effects: in particular, they tend to take resources away from the richest kingdoms and the
poorest kingdoms and toward the middle resource kingdoms.
We then take up poor rewarding contest success functions. Observe that, by definition, a
poor ruler gains from fighting a rich rival. However, in this setting, waiting is better: so the
poorer ruler would prefer to wait and allow for opponents to become large before engaging
in a fight. This gives rise to the prospect of peace. To make progress we divide the analysis
into two parts. To start, consider resource distributions with a single rich ruler: if this ruler
is sufficiently rich then his kingdom becomes an ‘irresistible’ prize; all other rulers have a
strict incentive to fight to acquire the rich kingdom. So peace cannot be sustained and the
outcome is hegemony. Next, consider the case where no ruler is very rich. Here we show
that perpetual peace and a phase of war followed by peace may be sustained in equilibrium.
The key to sustaining peace is the threat of imminent war. The equilibrium has the following
structure: no ruler wishes to fight a single fight because, once this fight is undertaken, all
rulers have an incentives to fight till the finish. It is this latter phase of war that makes
war today unattractive. These arguments are summarized in Proposition 4. We illustrate
through examples that the role of inequality is more general: across a range of networks and
in the setup where the neighborhood relation is complete and does not constrain the possible sequences offights.
3
contest success functions, peace is more likely when resources are more similar. And, we
illustrate through examples, that the dynamics of appropriation in the poor rewarding setting
are ‘equalizing’. This is most clearly seen when Tullock paramter γ is close to 0: across a
range of networks, the equilibrium payoffs of all rulers are then more or less equal.
To summarize, the analysis suggests that in the baseline model, rich rewarding technology
creates powerful incentives for war: starting in a situation with multiple kingdoms, the dy-
namics are characterized by incessant fighting; the expansion of a kingdom and, consequently,
the size of the empire, is limited by the connectivity of the network. By contrast, if the tech-
nology is poor rewarding, the dynamics are considerably more complicated. War followed by
hegemony is possible, but peace with multiple kingdoms may also be a long run outcome.5
The interest turns next to other factors that would potentially act as restraints on war.
We study the possibility of rivals forming an alliance to resist the aggression of an active
ruler. When a ruler is picked to fight, the other rulers can form an alliance: this alliance
puts together resources of all members to defend attack against any of them. The study of
such defensive alliances delineates the circumstances under which a ‘balance of power’ can
help restrain aggression and limit hegemony. Next we turn to costs of war in terms of lost
resources. We consider a model in which rulers lose a proportion of resources in war. The
analysis shows that costs of war significantly alter the incentives of rulers to wage war. A
richer ruler will only wish to attack a poorer ruler if the resource differences are neither too
small nor too large. This creates the possibility of ‘buffer’ states: poor kingdoms that are
located in between large powerful neighbors and prevent the progression of conflict. Alliances
and costs of war thus highlight the ways in which the framework can be enriched in ways that
accommodate forces that limit the scope of hegemony.
We now place our paper in the context of the literature and clarify its contributions. Our
paper studies the dynamics of war and peace and the formation of empires; related work
includes Hirshleifer [1995], Jordan [2006], Krainin and Wiseman [2016], Levine and Modica
[2013, 2016], and Piccione and Rubinstein [2007]. In an early paper, Hirshleifer [1995] showed
that ‘anarchy’ or multiple opponents could be sustained in a dynamic setting only if the
technology satisfies γ < 1.6 We show that γ > 1 does indeed lead to the emergence of
5We have also considered a number of other factors: ties in a war, the guns vs butter trade-off, resurrectionof defeated rulers, and asymmetric contest success functions. The details of these extensions are available fromthe authors.
6Fearon [1996] presents a result along similar lines: he considers a two-actor game in which states 1 and 2bargain over a territory. Let xt ∈ [0, 1] be the territory 1 controls at time t. Suppose p(xt) is the probabilitythat 1 prevails if the states fight at time t. Drawing an analogy with the present paper, this technology is
4
a hegemon, but our analysis goes beyond this insight along a number of dimensions: we
develop a non-cooperative and dynamic game with many far-sighted players; we consider
general contest functions, and there is a network structure which shapes the sequence of
attack strategies and the scale of empires. Our analysis introduces new concepts: rich/poor
rewarding contest success functions and strong/weak rulers. They enable us to address a
range of different questions, such as the timing and monotonicity of optimal attack strategies,
and how the prospects of individual rulers depend on the network and on the nature of the
contest success function.
The theoretical framework combines elements from the literature on contests, on resource
wars, and on networks. We now discuss the relationship between our paper and these litera-
tures.
There is a large literature on contests, for surveys see Konrad [2009] and Garfinkel and
Skaperdas [2012]. We consider a general model of multi-player contests inspired by the ax-
iomatic work of Skaperdas [1996].7 In recent work, Konrad and Kovenock [2009], Groh,
Moldovanu, Sela, and Sunde [2012], and Anbarcı, Cingiz, and Ismail [2018] study multi-player
sequential contests. In these papers the contest takes the form of an all-pay auction. The
interest is in how individual heterogeneity and the sequential contest structure determine ag-
gregate efforts and winning probabilities. By contrast, in our model, we abstract away from
effort so that we can study the dynamics of conflict with general contest success functions and
networks. To the best of our knowledge, the results on rich/poor rewarding contest success
functions and strong/weak rulers, and the mapping from these results to imperial history, are
novel in the context of this literature.8
The role of resources in shaping violent conflict is an active field of study, see e.g., Ace-
moglu, Golosov, Tsyvinski, and Yared [2012], Caselli, Morelli, and Rohner [2015], and Novta
[2016]. This literature provides evidence for appropriation of resources as a major motivation
for war. The theoretical work is mostly limited to two players or to symmetric models; for an
overview of the theory, see Baliga and Sjostrom [2012]. Our paper contributes to this literature
by studying the cumulative dynamics of appropriation and the expansion of territory within
rich-rewarding if p(x) ¿ x when x > 1/2. Fearon [1996] shows that the distribution of territory converges to ahegemon when p is rich-rewarding and to an even distribution of territory when p is poor-rewarding.
7For an early study of optimal strategy of attack in a three player game, see Shubik [1954]. Olszewski andSiegel [2016] study static contests with a large numbers of players.
8In our model, a rich rewarding contest success function provides a rationale for waging a sequence of warsdue to the compounding of spoils of war. This bears some resemblance to the earlier work of Garfinkel andSkaperdas [2000] and McBride and Skaperdas [2014] who study incentives for war in settings where rewardsextend through time. In their model, war today is attractive as it facilitates expansion tomorrow.
5
a contiguity network, and by linking these dynamics to major episodes of world history.
Finally, our paper is a contribution to the recent literature on conflict and networks, see
e.g., Franke and Ozturk [2015], Hiller [2017], Kovenock and Roberson [2012], Huremovic [2015],
Jackson and Nei [2015], and Konig, Rohner, Thoenig, and Zilibotti [2017]. For an overview see
Dziubinski, Goyal, and Vigier [2016]. Our paper advances this literature on two fronts: one,
the dynamics of appropriation in inter-connected conflict and two, how these dynamics are
decisively shaped by the contiguity network, the resources, and the contest success function.
The rest of the paper is organized as follows. Section 2 presents the basic model. Section 3
studies the incentives to fight and the optimal timing of attack. Section 4 presents the results
on equilibrium dynamics. Section 5 discusses two variants of a model: one where sequences
of attack are limited to a single fight and another one, where we allow for losses in war. We
conclude in Section 6. Proofs of all the results are moved to the Appendix.
2 The Model
We study a dynamic game in which rulers seek to maximize the resources they control by
waging war and capturing new territories. There are three building blocks in our model:
the interconnected ‘kingdoms’, the resource endowment for every kingdom, and the contest
success function.
Let V = {1, 2, . . . , n}, where n ≥ 2 is the set of vertices. Every vertex v ∈ V is endowed
with resources, rv ∈ R++. The vertices are connected in a network, represented by an undi-
rected graph G = 〈V,E〉, where E = {uv : u, v ∈ V, u 6= v} is the set of edges (or links)
in G. A network G is said to be connected if there is a path between any two vertices. For
expositional simplicity, we restrict attention to (undirected) connected networks. Our insights
extend in a natural way to directed networks.
A link between two vertices signifies ‘access’. Access may reflect physical contiguity. But,
in principle, it goes beyond geography: we do not restrict attention to planar graphs.9 So our
model allows for ‘virtual’ links, i.e., links made possible by advances in military and transport
technology.
Every vertex v ∈ V is owned by one ruler. At the beginning, there are N = {1, 2, . . . , n}rulers. Let o : V → N denote the ownership function. The resources of ruler i ∈ N under o,
9A graph is planar if it can be embedded in a plane, i.e. drawn in a plane in such a way that the edgesintersect at their endpoints only. An example of a graph that is not planar is a clique with 5 nodes.
6
are given by
Ri(o) =∑
v∈o−1(i)
rv (1)
The network together with the ownership configuration induces a neighbor relation between
the rulers: two rulers i, j ∈ N are neighbors in network G = 〈V,E〉 if there exists u ∈ V ,
owned by i, and v ∈ V , owned by j, such that uv ∈ E. Figure 1 illustrates vertices, resource
endowments, and connections; vertices controlled by the same ruler share a common colour.
The light line between vertices represents the interconnections, the dotted lines encircling
vertices owned by the same ruler indicate the ownership configuration, and the thick lines
between vertices reflect the induced neighborhood relation between rulers.
Figure 1: Neighboring Rulers
When two rulers fight, the probability of winning is specified by a contest success function.
Following Skaperdas [1996], we consider symmetric contest success functions with no ties.
Given two rulers, A and B, with resources xA ∈ R++ and xB ∈ R++, respectively, p(xA, xB)
is the probability that A wins the conflict and p(xB, xA) is the probability that B wins the
7
conflict.
The game takes place in discrete time: rounds are numbered t = 1, 2, 3 . . . . At the start of
a round, each of the rulers is picked with equal probability. The chosen ruler, (say) i, chooses
either to be peaceful or to attack one or more rulers, in a sequence, such that subsequent
opponents in the sequence are neighbours of the ruler when they are attacked. If a ruler
attacks a rival, he does so with all his current resources. If he chooses peace, one of the
remaining rulers is picked, again with equal probability, and asked to choose between war and
peace, and so forth, until one of the asked rulers chooses attack or all the ruler are asked
and choose peace. If no ruler chooses attack, the game ends. If the attacker loses one of the
attacks in the sequence or wins all the attacks in the sequence, the round ends.10 When two
rulers i and j fight, the winner takes over the entire kingdom of the loser (and also inherits the
boundaries, and hence the connections). This dynamic is illustrated in Figure 1: the orange
kingdom wins the war with the red kingdom and expands. This expansion brings it in contact
with new neighbors, the light and dark green kingdoms. The game ends when all rulers choose
to be peaceful (the case of a single surviving ruler is a special case, as there is no opponent
left to attack). Observe that, given these rules, the game ends after at most n− 1 rounds. It
may of course end earlier: this happens if all the rulers choose peace at a round.
The configuration of kingdoms and rulers – who is a neighbor of whom – is (potentially)
evolving over time. Given a set of vertices U ⊆ V , G[U ] = 〈U, {vu ∈ E : v, u ∈ U}〉 is the
subgraph of G restricted to vertices in U and links between them. The set of valid ownership
configurations, given graph G, is denoted by
O = {o ∈ NV : for all i ∈ N , G[o−1(i)] is connected}. (2)
As the graph is fixed, for simplicity, we omit it as an argument.
A state is a pair (o, P ), where P ⊆ N is the set of rulers who were picked prior to the
current mover and chose peace at o. Ruler i, picked at state (o, P ) ∈ O × 2N\{i}, chooses a
sequence of rulers to attack. A sequence σ is feasible at o in graph G if either σ is empty, or
if σ = j1, . . . , jk and for all 1 ≤ l < k, jl /∈ {i, j1, . . . , jl−1} and jl is a neighbor of one of the
rulers from {i, j1, . . . , jl−1} under o in G. A sequence σ is attacking if it is non-empty. Let
N∗ denote the set of all finite sequences over N (including the empty sequence). A strategy of
10De Jong, Ghiglino, and Goyal [2014] introduced a model of conflict with resources and a network: thekey difference is that conflict is imposed exogenously. Links are picked at random and rulers must fight. Bycontrast, in the present paper, the choice of waging a war or being at peace is the central object of study.
8
ruler i is a function si : O× 2N\{i} → N∗ such that for every ownership configuration, o ∈ O,
and every set of rulers, P ⊆ N \ {i}, si(o, P ) is feasible at o in G.11 Given ruler i ∈ N and
graph G, the set of strategies of i is denoted by Si; S =∏
i∈N Si denotes the set of strategy
profiles.
The probability that ruler 1 with resources R1 wins a sequence of conflicts with rulers with
resources R2, . . . , Rm, accumulating the resources of the losing opponents at each step of the
sequence is
pseq(R1, . . . , Rm) =m∏k=2
p
(k−1∑j=1
Rj, Rk
). (3)
Given o, a set of rulers, P , and a strategy profile s = (s1, s2, . . . , sn) ∈ S, the probability
that the game ends at o′, is given by F (o′ | s,o, P ). We shall sometimes refer to a final
ownership configuration as an outcome. An outcome is a hegemony if it involves a ruler who
owns all the nodes. The expected payoff to ruler i from strategy profile s ∈ S at state (o, P )
is:
Πi(s | o, P ) =∑o′∈O
F (o′ | s,o, P )Ri(o′). (4)
Every ruler seeks to maximize his expected payoff. The goals of rulers have been studied
extensively; for classical discussions see Hobbes [1651], Machiavelli [1992], and for more recent
work see Jackson and Morelli [2007].12
A strategy profile s ∈ S is a Markov perfect equilibrium of the game if and only if,
for every ruler i ∈ N , every strategy s′i ∈ Si, and every state, (o, P ) ∈ O × 2N\{i}, Πi(s |o, P ) ≥ Πi((s
′i, s−i) | o, P ). The game has finite horizon and there are no simultaneous moves.
Therefore an argument based on backward induction can be used to establish equilibrium
existence. In addition, all equilibria are payoff equivalent: expected utility at the beginning
of the game, for every player, is the same under every equilibrium. To see why, consider a
11Observe that the only feasible sequence for rulers who do not own any vertices, and for the ruler who ownsall vertices, is the empty sequence.
12We assume that ruler’s utility is linear in resources. Risk-averse and risk-loving preferences can easilybe accommodated. Suppose utility is given by u(x), with u(0) = 0, u′ > 0 and u′′ < 0. This means thatu(x+ y) < u(x) + u(y). Expected payoff to x vs y can be written as:
p(x, y)u(x+ y) = p(x, y)(u(x) + u(y))(1− d(x, y))
where d(x, y) = 1− u(x + y)/(u(x) + u(y)). So 0 < d(x, y) < 1: in other words, risk-aversion creates a ‘cost’of conflict.
9
state with only two active rulers, i and j. If i moves after j then i chooses between peace or
attacking j. The ruler is indifferent between these choice only if they are payoff equivalent.
Proceeding backwards, we can show that all equilibria yield the same expected payoff to all
players at all states.
Proposition 1. Fix a connected graph G. For any symmetric contest success function, p, and
any resource endowment, r ∈ RV++, there exists an equilibrium and all equilibria are payoff
equivalent.
3 The Incentives to Fight
This section introduces a general class of contest success functions and presents general results
on incentives to fight for the two and three ruler setting. Understanding these basic setting
allow us to establish how incentives to fight are shaped by the properties of the contest success
function.
The notions of rich and poor rewarding contest success functions are introduced and a
characterization is presented in terms of standard properties such as increasing and decreasing
returns. The interest then turns to the timing and order of optimal attacks: conditions on the
contest success functions are obtained under which rulers prefer to wait/not wait to attack.
In general, a contest success function is function q : R2++ → [0, 1]2. Following Skaperdas
[1996]), we consider three axioms for contest success functions, together with an additional,
fourth axioms, that substitutes independence of irrelevant alternatives axiom for the case of
bilateral contests.13
A1 For all (x1, x2) ∈ R2++, q1(x1, x2) + q2(x1, x2) = 1,
A2 For all i ∈ {1, 2} and j ∈ {1, 2} \ {i}, qi(xi, xj) is increasing in xi and decreasing in xj,
A3 For all (x1, x2) ∈ R2++, q1(x1, x2) = q2(x2, x1),
A4 For all i ∈ {1, 2} and (x1, x2, x3) ∈ R3, qi(x1, x2)qi(x2, x3)qi(x3, x1) = (1− qi(x1, x2))(1−qi(x2, x3))(1− qi(x3, x1)).
13Skaperdas [1996] proposes five axioms for contest success functions, the first three of them correspond toaxioms A1-3, the fourth, consistency axiom, is always satisfied in the case of two bilateral contests, and thefifth axiom, independence of irrelevant alternatives (IIR), applies to contests with at least three participants.Our axiom A4 replaces IIR. It is equivalent to IIR in the case of three or more sided conflict and replaces itin the case of two sided conflicts, where IIR has no bite.
10
By axiom A3, the contest success function is symmetric and can be represented by function
p : R2++ → [0, 1], where q1(x1, x2) = p(x1, x2) and q2(x1, x2) = p(x2, x1). Using the additional
axiom, A4, the proof of Skaperdas [1996] extends to show that a bilateral contest success
function satisfying axioms A1-4 necessarily takes the form
p(x, y) =f(x)
f(x) + f(y). (5)
with an increasing, positive, function f : R++ → R++.14 The study of contests remains a very
active field of study; see Fu and Pan [2015] for a a recent contribution and for references to
the literature.
Recall that (x + y)p(x, y) is the expected payoff of a ruler with resources x who fights an
opponent with resources y. We shall say that the contest success function, p, is rich rewarding
if for all x, y ∈ R++ with x > y,
(x+ y)p(x, y) > x (6)
Similarly, we shall say that p is poor rewarding if for all x, y ∈ R++ with x < y,
(x+ y)p(x, y) > x (7)
A rich rewarding contest success function gives the richer side an incentive to fight, while
poor rewarding one gives the poorer side an incentive to fight. We characterize rich and poor
rewarding contest success functions in terms of standard properties of the function f . We also
examine the timing of optimal attack: whether to attack now or to wait and attack later. A
contest success function, p, is said to have the no-waiting property if for all x, y, z ∈ R++,
p(x, y)p(x+ y, z) > p(x, y + z). It is said to have the waiting property if for all x, y, z ∈ R++,
p(x, y)p(x+y, z) < p(x, y+z). With contest success functions having the no-waiting property,
it is profitable for a ruler to attack the other two rulers in a sequence rather than wait to fight
the merged kingdom. The converse is true in the case of contest success functions that exhibit
the waiting property. Rich/poor rewarding and the timing of attacks are intimately related.
Turning to the optimal order of attack, a contest success function has the poor-first property
if the expected payoffs of attacking the poor ruler followed by the rich ruler are larger, i.e., for
all x, y, z ∈ R++, with y < z, p(x, y)p(x+y, z) > p(x, z)p(x+ z, y). A contest success function
has the rich-first property if the converse holds. Define
14In addition, f is unique up to positive multiplicative transformations.
11
h(s, t) =f(t)f(s+ t)
f(s+ t)− f(s)− f(t).
Proposition 2. Consider a contest success function, p, that satisfies (5). The function p is
rich rewarding if and only if f exhibits increasing returns to scale; it is poor rewarding if and
only if f exhibits decreasing returns to scale. In addition:
1. Timing of attack: If p is rich rewarding then it has the no-waiting property, while if p
is poor rewarding then it has the waiting property.
2. Order of attack: p has the poor-first (rich-first) property if and only if h(s, t) is strictly
increasing (decreasing) in t ∈ R++, for all s ∈ R++.
The argument for the first part proceeds as follows. Suppose that x > y. If f exhibits
increasing returns15 then f(x)/(f(x)+f(y)) > x/(x+y). Multiplying both sides by x+y now
yields the desired implication. On the other hand, if the stronger side gains in expectation,
then it must be that (x + y)f(x)/(f(x) + f(y)) > x. Rewriting and rearranging this gives
us the inequality f(x)/(f(x) + f(y)) > x/(x + y), which requires that f exhibits increasing
returns. A similar line of reasoning applies to the poor rewarding case. The argument for the
second part proceeds as follows. In the case of timing of attack, we begin by showing that the
no-waiting property is equivalent to f being super-additive. The next step demonstrates that
super-additivity is a weaker property than increasing returns to scale, and that concludes the
proof. In the case of order of attack, rewriting of the poor-first property derives the required
expression.
We note that the optimal order of attack result can be generalized to cover n opponents: if
all opponents are neighbours, then the order of attack is monotonically increasing (decreasing)
in the resources of opponents if h(x, y) is increasing (decreasing) in y for all x (this result is
stated and proved in the Appendix).16
15Function f exhibits increasing returns to scale if and only if f(x)/x is increasing on its domain and itexhibits decreasing returns to scale if and only if f(x)/x is decreasing on its domain.
16The qualification ‘if all opponents are neighbors’ is important. If some opponents are not neighbors thenit may be optimal to attack a richer neighbor in preference to a poor neighbor, so as to reach other pooreropponents first. Here is an example. Suppose G is a line network with 4 rulers, a, b, c, and d, each controllingone vertex (in that order). Suppose that resources of ruler a are x ∈ (0, 2). The resources of b, c and d arerespectively 2, 2.01 and 1. Assume Tullock contest success function with f(x) = x2. If x < 1.83 then theoptimal full attacking sequence of ruler b is (a, c, d): so it prescribes attacking the weakest neighbor first. Onthe other hand, if x > 1.84 then the optimal full attacking sequence is (c, d, a): it is better to first attack astronger neighbor, c, to get access to weak d, and only then attack a.
12
We illustrate the scope of these results through a consideration of the widely studied
Tullock contest success function.17
p(x, y) =xγ
xγ + yγ,
where γ > 0. Hence, f(x) = xγ. If γ > 1 then f has increasing returns to scale. From
Proposition 2 it follows that the contest success function is rich rewarding and has the no-
waiting property. On the other hand, if γ < 1, then f exhibits diminishing returns to scale.
It is therefore poor rewarding and the ruler would prefer to wait. Finally, observe that
(x + y)p(x, y) = x, for all x, y ∈ R++ if γ = 1. So the contest success function is reward
neutral ; it is also timing neutral (as for all x, y, z ∈ R++, p(x, y)p(x + y, z) = p(x + y, z)).
Lastly, in the case of γ > 1, h(s, t) is increasing in t for all s. Hence in this case the contest
success function has the poor first property. On the other hand, in the case of γ < 1, h(s, t) is
decreasing in t for all s and the contest success function has rich first property. Since h(s, t)
remains constant in t for all s, if γ = 1, so the contest success function is order neutral in this
case. To summarize:
Corollary 1. The Tullock contest success function is rich rewarding, has the no-waiting and
poor first properties if γ > 1; it is poor rewarding, has the waiting and rich first properties if
γ < 1. It is reward, timing, and order neutral if γ = 1.
The condition with regard to order of attack generalizes to larger sequences. Hence, in the
case of the Tullock Contest Function, it yields a clean implication: if γ > 1 then the optimal
attack strategy prescribes attacking rivals in increasing order of resources; the converse holds
if γ < 1. These results set the stage for the study of n ≥ 3 rulers located in a connected
network.18
We have not been able to locate clear empirical evidence on the nature of contest success
functions. The key of resources has been noted in the context of the formation of the first
Chinese Empire and the expansion of the Roman Empire (Lewis [2010], Polybius [2010]). For
more recent times, Clausewitz [1993], drawing inspiration from the Napoleonic wars in Europe,
argued that superiority in numbers was fundamental: an army twice as large as its opponent
almost never lost the battle (not even against a great general like Bonaparte). Howard [2009]
17In the Appendix we disucss Hirshleifer Difference Contest Function.18The literature has tended to assume γ ≤ 1. This is because of concerns about the existence of an
equilibrium in models where resources are costly. In our setting, the ruler chooses whether to fight or not andin this setting the existence of equilibrium does not depend on the value of γ.
13
likewise argues that army size was critical factor in the victory of Germany over France in the
Franco-Prussian Wars. These observations are consistent with a probability of winning that
is responsive to army size and resources. In what follows, we therefore present equilibrium
analysis for both rich and poor rewarding contest success functions.19
4 Conquest and Empire
This section studies equilibrium dynamics of war and peace and the formation of empires.
For the rich rewarding case, we show that equilibrium is characterized by incessant warfare
and that the outcome is hegemony. The connectivity of the network defines the limits of
the hegemony. Depending on the resources and location in the network, the rulers can be
characterized as either weak or strong. This characterization plays a key role in the analysis.
The analysis of poor rewarding contest functions is more partial because the dynamics are
considerably more complicated: we show that perpetual peace, perpetual war (and hegemony),
and a phase of war followed by peace can all arise in equilibrium.
Given ownership configuration o, the set of active rulers at o is
Act(o) = {i ∈ N : ∅ ( o−1(i) ( V }.
An ordering of the elements of the set Act(o) \ {i}, σ, such that the sequence σ is feasible
for i in G under o is called a full attacking sequence (or f.a.s). Figure 2 illustrates such a
sequence(for the orange kingdom).
We are now ready to state our first main result on equilibrium dynamics.
Theorem 1. Consider a rich rewarding contest success function that satisfies (5). Suppose
G is a connected network and let r ∈ RV++ be a generic resource profile. In equilibrium, every
active ruler chooses to attack a neighbor if |Act(o)| ≥ 3, and at least one of the active rulers
attacks his opponent if |Act(o)| = 2. The outcome is hegemony and the probability of becoming
a hegemon is unique for every ruler.
The result offers an account of the dynamics of conflict in a rich rewarding setting when
rulers are driven by a desire to maximize resources under their control. It predicts incessant
19To get a sense of the numbers, consider the Tullock Contest Function and suppose one army is twice thesize of the other army. With an exponent γ = 2, the probability of winning for the larger army is 0.8, andwith an exponent γ = 4 it is (approx) 0.95. On the other hand, with γ = 0.5 the probability of winning is 0.6,and with γ = 0 the probability is 0.5.
14
Figure 2: Full Attacking Sequence
fighting, preemptive attacks, and long attacking sequences. In particular, at every round
with at least 3 active rulers, every ruler (even a weak one) chooses to attack a neighbour in
equilibrium. It is worth drawing attention to the generality of this result: it holds for all rich
rewarding contest functions, for any connected network, and for generic resources.
We discuss the arguments underlying the theorem. A ruler is said to be strong if he has
an attacking sequence σ = i1, . . . , ik, where for all l ∈ {1, . . . , k},
l−1∑j=0
Rij(o) > Ril(o).
In other words, at every step in the attacking sequence, the ruler has more resources than
the next opponent. The set of strong rulers at ownership configuration o is
S(o) = {i ∈ Act(o) : i has a strong f.a.s. σ at o}.
A ruler who is not strong is said to be weak. Note that (generically) in any state, the ruler
with the most resources is strong, while the ruler with the least resources is weak. Thus both
sets are non-empty in every network and for (generic) resource profiles.
The first step is to show that, assuming that all other rulers choose peace in all states,
it is optimal for a strong ruler to choose a full attacking sequence. This is true because the
contest success function is rich rewarding and so a strong ruler has a full attacking sequence
that increases his resources in expectation, at every step, along the sequence. The second
step extends the argument to cover opponents who choose war. If opponents are active then
15
the no-waiting property (from Proposition 2) tells us that it is even more attractive to not
give them an opportunity to move. For a strong ruler it is therefore a dominant strategy to
use an optimal full attacking sequence. The final step in the proof covers non-strong rulers
to establish that with 3 or more active rulers, it is optimal for every ruler to choose a full
attacking sequence. Observe that we have already shown that every non-strong ruler knows
that he will be facing an attack sooner or later. This means that waiting can only mean that
the opposition will become (larger and) richer. The no-waiting property then tells us that
every ruler must attack as soon as possible. If there are only two active rulers then the richer
ruler has a strict incentive to attack the poorer opponent (this follows from the definition of
the rich rewarding contest function).
We now examine the role of the contiguity network and resources more closely. For ex-
positional simplicity, we focus on the Tullock contest success function. Notice that, due to
timing and order neutrality, there are no interesting network effects when γ = 1: equilibrium
expected resources of any ruler remain equal to his initial resources. When γ is large it is
never optimal to attack a richer ruler if other options are available. The optimal strategy for a
strong ruler must involve attacking a poorer ruler at every stage in the attack sequence. Such
a sequence is clearly not available for a weak ruler: the probability of a weak ruler becoming
a hegemon converges to zero, as γ grows.
Given the initial ownership configuration o0, a γ, and resources r, let Probi(r, γ | o0) be
the equilibrium probability of ruler i becoming the hegemon. Define
Prob∗i (r | o0) = Probi(r, limγ→+∞
γ | o0).
Proposition 3. Suppose the contest success function is Tullock, the network G is connected,
and the resources r ∈ Rn++ are generic. The probability of a weak player becoming a hegemon
becomes negligible as γ grows. Specifically,
Prob∗i (r | o0)
{≥ 1|Act(o)| , if i ∈ S(o0)
= 0, otherwise.
Whether a ruler is strong or weak depends both on the distribution of resources and on
the position of the ruler in the contiguity network. In Figure 3 we represent strong rulers in
red and weak rulers in yellow. It is helpful to define the boundary of a set of vertices U ⊆ V
in G is
BG(U) = {v ∈ V \ U : there exists u ∈ U s.t. uv ∈ E}
16
A set of vertices, U , is weak if G[U ] is connected, BG(U) 6= ∅, and for all v ∈ BG(U),
rv >∑
u∈U ru. A weak set of nodes is surrounded by a boundary, constituted of nodes, each
of whom is endowed with more resources than the sum of resources of vertices within the set.
Weak sets are illustrated in Figure 3. It is easy to see that, for any initial state o, a ruler is
weak if his vertex belongs to a weak set and, otherwise, the ruler is strong.
Figure 3: Weak rulers (surrounded by thick lines) and strong rulers
Proposition 3 covers the case of large γ. We now turn to examples to show that the
distinction between strong and weak rulers is central to the study of dynamics more generally,
across rich rewarding γ. Consider three networks with 10 nodes: the clique network (with 45
links), a connected network with 27 links and a tree network (with 9 links). The resources
endowments at the nodes are 2, 3, 6, 11, 13, 15, 16, 18, 21, and 23, respectively. The strong
rulers are presented in purple, while the weak rulers are presented in yellow. These networks
and resource endowments are presented in Figure 4. Observe that as we delete links from
clique to obtain the network with 27 links, the number of weak rulers increases strictly (from
2 to 3) and the same happens as we go move from network with 27 links to the network with
9 links (the number goes up from 3 to 4).
We compute the equilibrium payoffs in these examples;20 the results are summarized in
Figure 5.21 The key point to note is that, even for γ = 8, the long run prospects of a ruler are
20We would like to stress that all the computational examples in the paper are obtained by means ofnumerical calculations of equilibrium strategies and payoffs and not by simulations. This allows us to obtainmuch more accurate results.
21In the figures we present the relation between initial end expected equilibrium resources using scatterdiagrams and we present the distribution of resources using Lorenz curves. For any x ∈ [0, 100], a Lorenzcurve represents the fraction of the total resources owned by poorest x% of the rulers.
17
essentially determined by whether he is strong or weak. Further study of examples that span
a range of different values of γ reveal that this pattern is reinforced when we increase γ.
Figure 4: Examples of Networks
Given the importance of strong and weak rulers, we briefly comment on how changes in
resources and links affect the set of strong and weak rulers. Given a resource profile, observe
that adding links to a network offers all rulers potentially more sequences of attack. This
means that a ruler who was weak may now have a strong sequence. Adding links (weakly)
therefore expands the set of strong rulers. The number of strong rulers is maximized in the
complete network and it is minimized when the strongest ruler is at the center of a star
network. Given a network and a resource configuration, an increase in resources of a ruler
either maintains his status or switches him from weak to strong. Observe that an increase in
resources of a ruler may well lead to another ruler becoming weak. From Proposition 3 we
can infer that additional resources for one ruler can make a big difference to his and others’
long term prospects.
The discussion so far has focused on the difference between strong and weak rulers. We
now argue that the network structure also shapes the relative prospects of different strong
rulers. Consider an example with two strong rulers. Suppose the two rulers are 1 and 2, and
they own vertices v1 and v2, respectively. The set of the remaining vertices, V \ {v1, v2}, can
be partitioned into three sets: the set of nodes reachable from v2 via v1 only, denoted by U1,
the set of nodes reachable from v1 via v2 only, denoted by U2, and the remaining nodes, U12
(c.f. Figure 6).
To see the effects of the networks structure easily, suppose that γ is large and that rv1 +
RU1 > rv2 + RU2 + RU12 . This ensures that ruler 1 remains strong as long as he is active.
Ruler 2, on the other hand, becomes weak if ruler 1 accumulates enough resources from the
18
00 5 10 15 20 2500
5
10
15
20
25
Initial Resources
Equilib
rium
Pay
offs
Clique 27 linksTree
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of Rulers (Cumulative)
Fra
ctio
nof
Eq.
Pay
offs
(Cum
.)
Clique 27 linksTree Initial
Figure 5: Equilibrium Payoffs and Lorenz Curves: γ = 8.
set U1. The probability of ruler 1 becoming a hegemon is approximately 1/2 + q, where q is
the probability that ruler 2 is picked to move before ruler 1 is picked to move and he is weak
when that happens. Thus q is the probability that 1 conquers sufficiently many nodes before
ruler 2 is picked to move. To fix ideas, suppose that 1 needs to acquire all the nodes in U1 to
become uniquely strong. Suppose |U1| = k. If G[U1] is a fully disconnected network then q
is approximately equal to k!/(k + 2)! = 1/((k + 1)(k + 2)). If, on the other hand, G[U1] is a
clique with k − 1 strong rulers then q is approximately equal to (k − 1)(k + 1)!/(2(k + 2)!) =
(k−1)/(2(k+2)). As k gets large, the probability that ruler 1 becomes the hegemon converges
to 1/2 in the former case, and to 1 in the latter case.
Figure 6: Partitioning of a Graph with Two Strong Rulers
4.1 Poor Rewarding Contest Success Functions
Recall that in the poor rewarding setting, every bilateral conflict is profitable to the poorer
of the two opponents. However, the poor rewarding property also implies that rulers have a
preference to wait before they fight. These two considerations suggest that the dynamics can
be complicated. We are especially interested in the possibility of peace.
19
We start by noting that in equilibrium, at every ownership configuration, there is either
peace or fight, regardless of the order in which the rulers are picked to move. Formally, given
a strategy profile, s, an ownership configuration o ∈ O is peaceful under s, if for all i ∈ Nand all P ∈ 2N\{i}, si(o, P ) is the empty sequence. An ownership configuration o ∈ O is
conflictual under s if for every sequence i1, . . . , in of rulers from N there exists k ∈ {1, . . . , n}such that sik(o, {i1, . . . , ik−1}) is not empty. In other words, regardless of the order in which
the rulers are picked to move at o, one of the rulers chooses an attacking sequence.
By the observation above, the possibility of peace means that, in equilibrium, there exist
ownership configurations, with two or more active rulers, at which all the rulers prefer staying
peaceful to choosing fight. To make progress we divide the analysis into two parts: first, we
characterize situations where peace is impossible, and second, we turn to situations where
peace may be sustainable. We say that there is perpetual peace in a given strategy profile, if
the initial state is peaceful. We say that there is war followed by peace in a given strategy
profile if the initial state is not peaceful and no equilibrium outcome is hegemony.
Proposition 4. Consider a generic poor rewarding contest success function that satisfies (5).
1. For any connected network, G, and any generic resource endowment, r ∈ RV++, every
ownership configuration o ∈ O is either peaceful or conflictual in equilibrium.
2. For any connected network, G, any node v ∈ V , and any resource endowment of the
other nodes, r−v, there exists a resource level rv such that for all rv > rv, there is fight
till hegemony in equilibrium under resource endowment (rv, r−v).
3. For any n ≥ 4, there exists a network and a generic resource profile such that there is
perpetual peace in equilibrium. Similarly, there exists a network and a generic resource
profile such that there is war followed by peace in equilibrium.
The result should be seen as a possibility result: it illustrates the rich range of outcomes
possible under the poor rewarding contest success function. A comparison of Theorem 1 with
Proposition 4 reveals contrasting optimal strategies (full attacking sequence versus no fighting)
and outcomes (hegemony versus multiple kingdoms) and highlights the key role of the contest
success function in shaping conflict dynamics. The hegemony result relies on quite different
arguments than the hegemony result under rich rewarding contest success function. In the
poor rewarding case, the existence of a sufficiently rich ruler motivates other rulers to fight.
However, due to the waiting property, these rulers may choose to fight only if others do not.
20
This is in contrast to the rich rewarding case, where each ruler chooses fight whenever he is
given a chance. The peace and war followed by peace outcomes rely on the idea of fear of
conflict escalation. We propose a network and a (generic) resource profile for which, whenever
any ruler chooses to fight, there will be fight till hegemony in the following states and the ruler
who started the conflict will be involved in all the following conflicts. The resource endowments
are such that it is never profitable for any ruler to be involved in fight till hegemony starting
from the initial state. The main challenge is to show that such a resource endowment exists,
for general n.
We next examine how networks, resources, and the contest success function affect the
prospects of peace. We consider Tullock contest success function with two values of γ: 0.05
(low) and 0.8 (high) and networks with 10 nodes (as in the rich rewarding case). In addition we
consider eight ranges of resources: [45, 55], [40, 60], [35, 65], [30, 70], [25, 75], [20, 80], [15, 85],
[10, 90]. For each triple of γ, number of links, k, and resource range, [a, b], we pick 1000
random samples of connected networks of k links with resources drawn uniformly from the
set (of 10, 000 evenly spaced values from) [a, b]. Figure 7 presents the frequencies of samples
exhibiting peace in the first round as a function of the resource range. It suggests that peace
is more likely when resources are drawn from a smaller range: this is true for both high and
low values of γ and true also across a wide range of networks. Taking together, Proposition 4
and our examples show that resource equality is conducive for peace.
0 10 20 30 40 50 60 70 80 900
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Resource Range
Frequency
ofPeace
Clique 27 linksTree
0 10 20 30 40 50 60 70 80 900
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Resource Range
Frequency
ofPeace
Clique 27 linksTree
Figure 7: Frequency of Peace: γ = 0.05 (left), γ = 0.8 (right).
This section concludes with an observation on equilibrium payoffs. We take up the same
three networks as in the rich rewarding case (from Figure 4) and we fix the Tullock parameter
γ to be equal to 0.05. Figure 8 presents the equilibrium payoffs and the Lorenz curves for the
three networks and the initial resources. It is clear that, when γ is very small, the equilibrium
21
dynamics are powerfully equalizing. A comparison with Figure 5 also reveals the big difference
between the rich and poor rewarding setting: the poorer kingdoms gain significantly in the
latter setting, and this is reflected at the aggregate level via the Lorenz curves.
00 5 10 15 20 2500
5
10
15
20
25
Initial Resources
Equilib
rium
Pay
offs
Clique 27 linksTree
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of Rulers (Cumulative)
Fra
ctio
nof
Eq.
Pay
offs
(Cum
.)
Clique 27 linksTree Initial
Figure 8: Equilibrium Payoffs and Lorenz Curves: γ = 0.05.
5 Extensions
In this section we consider two extensions of the benchmark model. Firstly, we consider a
variant where the attack sequence chosen by the rulers at each round consist of only one
opponent. This allows the rulers to react quicker to others moving. Second, we study the
potential of alliances to block the spread of a kingdom. Thirdly, we consider a variant where
each fight results on losses to resources. This reduces incentives of the rulers to choose war
and, if the losses in war are sufficiently large, allows for peace to emerge.
5.1 Short Attack Sequences
In the basic model, a ruler is allowed to choose a full attacking sequence of attacks. In
particular, all other rivals remain passive, while this ruler executes this sequence. In this
extension, we allow for rivals to have more opportunity to react and the goal of this section
is to examine if our results are robust to this generalization.
We consider a variant of our model where rulers, when picked to move, can either choose
peace or choose a sequence of attack of length 1 only, and then a new mover is drawn. A
strategy of a ruler i is a function si : O × 2N\{i} → N ∪ {ε} such that for every ownership
configuration, o ∈ O, and every set of rulers, P ⊆ N \ {i}, si(o, P ) is feasible at o in G, that
22
is either si(o, P ) is empty or si(o, P ) consists of a neighbor of i under o in G. As the problem
is especially relevant under the no-waiting property, in this discussion, we restrict attention
to rich rewarding contest success functions. Notice that the proof of Proposition 1 can be
adjusted in a straightforward way and so Proposition 1 is valid for the short-attack variant
of the model. In particular, equilibrium existence and payoff equivalence of equilibria hold in
this model as well.
First, we take up the setting with a unique strong ruler. This situation arises naturally
if one ruler controls more than half of the resources. But the condition is significantly more
general. Given any network, G, recall that a maximal set of nodes such that any two distinct
nodes in the set are reachable from each other by a path in G is called a component in G.
The set of all components of G is denoted by C(G). In addition, given a set of nodes, U ⊆ V ,
G−U = G[V \U ] denotes the graph obtained by removing the nodes in U and all their links
from G. A connected graph G with resource endowment r has a unique strong node if and
only if there exists a node v ∈ V such that for every component C ∈ C(G− {v}), rv > RC .
Proposition 5. Consider a rich rewarding contest success function that satisfies (5). Suppose
the network G is connected and a (generic) resource profile r ∈ Rn++ is such that there is exactly
one strong node. In equilibrium, at every ownership configuration, o, at least one ruler attacks
his neighbor. So the outcome is hegemony and the probability of becoming a hegemon is unique
for every ruler.
The proof is presented in the Appendix. The first observation is that if there is a unique
strong ruler under some ownership configuration, then in every ownership configuration that
follows in the course of the game, there is also a unique strong ruler. This is because no
weak ruler can become strong, unless he fights and beats a strong ruler (in which case he
becomes the unique strong ruler). Given this observation, we now show that at any state,
for any strategy profile of the other rulers, the unique strong ruler increases his resources in
expectation using the ‘optimal attacking’ strategy. We proceed by induction. For two rulers,
which is the base step, the claim clearly holds. Assume now that the claim holds for k rulers.
We show that the result holds for k+1 active rulers. This is because, due to the rich-rewarding
contest success function, any fight between the strong ruler and any other ruler increases his
resources in expectation and then, by the induction hypothesis, the expected resources at
the end of the game are even higher. If any other two rulers fight, then the resources of the
strong ruler remain unchanged in any following state and then, by the induction hypothesis,
they increase. Thus, in any equilibrium there cannot be peace, because, at any ownership
23
configuration, the strong ruler prefers to attack one of his neighbours over remaining peaceful.
00 5 10 15 20 2500
5
10
15
20
25
30
35
40
Initial Resources
Equilib
rium
Pay
offs
Clique 27 linksTree
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of Rulers (Cumulative)
Fra
ctio
nof
Eq.
Pay
offs
(Cum
.)
Clique 27 linksTree Initial
00 5 10 15 20 2500
5
10
15
20
25
Initial Resources
Equilib
rium
Pay
offs
Clique 27 linksTree
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of Rulers (Cumulative)
Fra
ctio
nof
Eq.
Pay
offs
(Cum
.)
Clique 27 linksTree Initial
Figure 9: Equilibrium Outcomes γ = 8: short attacks model (top) and basic model (bottom).
We now examine the case of multiple strong rulers, with the help of examples. Consider
the same three networks (with corresponding resources) as in the basic model (c.f. Figure 4)
and assume that γ = 8. Equilibrium payoffs and Lorenz curves for the results are presented
in Figure 9. By way of illustration, the figure contains also the corresponding outcomes for
the basic model. In all the examples every ruler chooses to fight in every state and so the
outcome is hegemony. As in the case of the basic model, the expected resources of the weak
rulers are close to 0. There is, however, much greater variation across the strong rulers.
Their equilibrium payoffs are more affected by the initial resource distribution, as compared
to the basic model. The ‘richest’ ruler gains most from the dynamics and has much higher
expected payoffs. The Lorenz curves confirm this point: the one-step dynamics lead to greater
inequality than the dynamics in the basic model.
Next, we study the frequency of peace. Suppose again that n = 10 nodes. We run
24
numerical calculations for γ ∈ {2, 4, 8, 16, 32}, number of links, k ∈ {9, 18, 27, 36, 45}, and
resource ranges [45, 55], [40, 60], [35, 65], [30, 70], [25, 75], [20, 80], [15, 85], and [10, 90]. For
each combination of the three parameters we have drawn 1000 random samples, ensuring that
there are at least two strong rulers. In each case we observe that there is fight till hegemony
in equilibrium. Moreover, at every state (on and off the equilibrium path) all rulers chose
fight. Taken together, Proposition 5 and these examples suggest that incessant warfare and
the emergence of hegemony are robust features of the dynamics of appropriation in the rich
rewarding setting.
5.2 Alliances
In the basic model an individual ruler chooses to attack a sequence of other rulers, one at a
time. The background assumption is that the rulers that are being attacked must confront
the attacker on their own. In the face of a powerful attacker, it would be reasonable for a
ruler to form an alliance with other rulers who may eventually have to also face the same
attacker. Starting with the classical account of Thucydides, alliances are a recurring theme
in the history of warfare and empires. They have also been the subject of recent work, see
Bloch [2012], Konig et al. [2017] and Jackson and Nei [2015]. An analysis of alliances is
important but it is outside the scope of the present paper. In this section, our goal is more
limited: we wish to elaborate on some considerations – relating to resources and networks
– that arise when rulers can form such defensive alliances. We leave a more systematic and
general analysis to future work.
We consider the following slight variation on our model: in a round, once a ruler has been
picked, all the other active rulers have an opportunity to create alliances. The function of an
alliance is limited: it puts together the resources of all its members and these resources can be
deployed to defend any member of the alliance against an attack. To develop some intuition
for how this can affect the dynamics of war suppose that the contest function is Tullock and
that γ is large. It follows then that a ruler i will not want to attack a neighbor j who is part
of an alliance that has more resources.
An important question that arises at this point is who can form alliances with whom.
In the simplest case, suppose there is no restriction. In other words, if i is picked all other
active rulers can contemplate an alliance. In this case a ruler i will attack only if he has
more resources than the sum of all the resources of the other rulers. This leads to our first
observation: when alliances are unrestricted, hegemony will arise if and only if there is a ruler
25
who controls more than half of all resources.
Next, we consider a simple restriction on alliance membership: suppose that members of an
alliance must constitute a connected sub-graph of the residual contiguity network involving all
rulers other than the ruler currently picked. Observe that in the case of a complete network
all alliances are feasible and this formulation then corresponds to the simpler unrestricted
setting considered above. However, in more general networks, even a relatively poor ruler can
attack neighbors and become a hegemon. As an example consider a line network with an odd
number of rulers: the central node has (n + 1)/2 units of resources, while each of the other
n − 1 nodes has exactly 1 unit of resources. The central node controls roughly one third of
all resources but there is no defensive alliance that can successfully protect the neighbors of
the central node. For γ large, it is optimal for the central node to implement a fully attacking
sequence. To rule out such situations we develop a sufficient condition for absence of war (and
consequently the lack of hegemony). Consider an ownership configuration with three or more
active rulers. There is no war in this configuration if for every ruler, i, it is the case that each
of his neighbors is part of an alliance whose resources add up to more than the resources of
ruler i.
These observations provide a theoretical basis for the notion of ‘balance of power’; for an
early discussion of this concept, see Hume [2006]; for more recent explorations of the idea, see
Kissinger [2015] and Betts [2013]. ‘Balance of power’ offers an alternative foundation to order
and stands in contrast to a theory in which order arises out of hegemony.
5.3 Losses in War
In the basic model, the winner retains all his resources and captures the entire resources of the
loser. Wars entail destruction of infrastructure and loss of lives; these losses can be especially
high in case the rivals use nuclear bombs. When losses are small, our earlier arguments
continue to hold, while if losses are very large then no ruler has an incentive to fight. The
interesting case is therefore one where the losses take on an intermediate value. We will show
that this intermediate range can give rise to the phenomenon of buffer states.22
Let p be a rich rewarding contest success function. Suppose that a conflict between two
rulers entails a loss of a fixed fraction δ ∈ (0, 1) of total resources. The expected payoff to a
22In our basic model, for simplicity, we assumed that kingdoms have no trade ties. Such ties may have abearing on losses from a war. But the relation between trade and incentives to fight is not straightforward;for recent work on this question, see Jackson and Nei [2015] and Martin et al. [2008].
26
ruler with x resources from a conflict with a ruler with y resources is:
Π(x, y) = p(x, y)(x+ y)(1− δ).
We start by noting that the arguments in the proof of Proposition 1 can be extended to
establish equilibrium existence and uniqueness of equilibrium payoffs.
When δ = 0, there are no losses in war: this is the benchmark model. More generally,
regardless of the value of δ, with a rich rewarding contest success function, expected resources
of the poorer ruler are always smaller than his initial resources. Similarly, with a poor re-
warding contest success function, expected resources of the richer ruler are always smaller
than his initial resources. The principal impact of losses in war is to discourage war: so, in
what follows, we focus on the rich rewarding case, as this is the case where all rulers have an
incentive to fight at all points in the basic model.
If δ > 1/2 then each ruler loses half of his resources in war: so it is clearly not profitable
for the richer ruler to fight. Hence we restrict attention to the case where δ < 1/2. Losses
in war have an interesting implication for the incentives to fight. To see this suppose x > y.
If (1 − δ)y < δx then the potential gain from the war is smaller than the loss to the richer
ruler, and so attacking very small opponents is not profitable for the rich ruler. On the other
hand, for a given δ < 1/2, the richer ruler has no incentive to attack the poorer ruler if y
is sufficiently close to x: the total resources from such an attack are less than 2x while the
probability of winning is less than 1/(2(1 − δ)) if y is sufficiently close to x. Thus losses in
war create an interesting structure of incentives: the richer ruler wishes to attack the poorer
ruler only if the ruler is neither too poor not too rich.
This structure of incentives gives rise to the possibility of buffer states. A buffer state is
a ruler, lying between greater powers, and preventing them from attacking each other. The
following example brings out this possibility.
Example 1 (Buffer State). Consider a line with three vertices (and rulers): a, b and c (in
that order). Suppose that Ra = 10 and Rc = 9, γ = 16 and the cost of conflict is δ = 0.2. Let
us look at incentives to wage war as we vary the resources of ruler b. If Rb ∈ (0, 1.79), ruler a
does not find it profitable to attack b and then c, nor attack b only. Similarly, c does not find
attacking b or attacking b and then a profitable. Therefore peace in the initial state is the
equilibrium outcome. Next consider Rb ∈ (1.79, 10): one of the rulers now has an incentive
to attack, and the outcome is either two rulers or a single hegemon. Finally, observe that if
there was a link between a and c, then ruler a would definitely attack c. So the poor kingdom
27
must offer the only path between the two large rulers. 4
More generally, a ruler i is a buffer state at ownership configuration o, if G−o−1(i) is not
connected and for any neighbour of i, Ri(o) < Rj(o). In other words, the vertices owned by
i fragment the network and the resources of the buffer state are smaller than the resources of
each of the surrounding rulers. The key step is to show that attacking a small ruler becomes
unprofitable not only in bilateral fights but also in longer sequences of fights. We show this
under a mild assumption that f ′(0) = 0. With this assumption the expected payoff from
a sequence of fights is increasing in resources of an intermediate opponent in the sequence
when these resources are sufficiently small and so, decreasing resource in this range decreases
the payoffs which, with sufficiently large losses in war, makes including the poor ruler in the
sequence unprofitable. These observations are formally stated in the proposition below.
Proposition 6. Fix a connected network G, resource endowment r ∈ RV++, and a rich reward-
ing contest success function, p, satisfying (5) with continuous and continuously differentiable
function f . Then there exist threshold values of the losses in war 0 < δ1 < δ2 ≤ 1/2 such that
1. If δ < δ1 then Theorem 1 holds.
2. If δ > δ2 then the equilibrium outcome is perpetual peace.
3. If f ′(0) = 0 then buffer states can help prevent a hegemony.
We conclude by noting that losses in war are a different force driving peace as compared to
the purely strategic considerations that arise under poor rewarding contest success function.
These two forces may, in fact, work against each other. Higher losses in war may prevent
conflict escalation and this may encourage rulers to attack their neighbors. An example
illustrating this idea is given in the Appendix.
6 Concluding Remarks
This paper develops a theoretical framework for the study of the incentives to wage war to
conquer territory and resources. Our innovation is that we locate the dynamics of appropria-
tion within a contiguity network. The analysis develops a number of results on the interplay
between the technology of war, the resources of rulers, costs of war, alliances, and contiguity,
that illuminate the process of the formation of empires.
28
References
D. Acemoglu, M. Golosov, A. Tsyvinski, and P. Yared. A dynamic theory of resource wars.
Quarterly Journal of Economics, 127:283–331, 2012.
N. Anbarcı, K. Cingiz, and M. Ismail. Multi-battle n-player dynamic contests. Working paper,
Maastricht, 2018.
S. Baliga and T. Sjostrom. The Hobbesian trap. In The Oxford Handbook of the Economics
of Peace and Conflict, Oxford Handbook in Economics. Garfinkel, M. and Skaperdas, S.,
New York, USA, 2012.
R. Betts. Conflict After the Cold War: Arguments on Causes of War and Peace. Pearson:
New Jersey, 2013.
F. Bloch. Endogenous formation of alliances in conflicts. In The Oxford Handbook of the Eco-
nomics of Peace and Conflict, Oxford Handbook in Economics. Garfinkel, M. and Skaperdas,
S., New York, USA, 2012.
F. Braudel. The Mediterranean and the Mediterranean World in the Age of Philip II: Volume
1. The University of California Press, London, 1995.
F. Caselli, M. Morelli, and D. Rohner. The geography of inter-state resource wars. Quarterly
Journal of Economics, 130:267–315, 2015.
C. Clausewitz. On War. Everyman Library, New York, 1993.
J. Darwin. After TamerLane: The Rise and Fall of Global Empires 1400–2000. Penguin.
London., 2007.
M. De Jong, A. Ghiglino, and S. Goyal. Resources, conflict and empire. mimeo, 2014.
M. Dziubinski, S. Goyal, and A. Vigier. Conflict and networks. In Y. Bramoulle, A. Gale-
otti, and G. Rogers, editors, The Oxford Handbook of the Economics of Networks, Oxford
Handbooks, pages 215–243. Oxford University Press, New York, US, 2016.
J. H. Elliott. Empires of the Atlantic World: Britain and Spain in America 1492–1830. Yale
University Press, New Haven, 2006.
29
J. Fearon. Bargaining over objects that influence future bargianing power. Working Paper,
University of Chicago, 1996.
J. Franke and T. Ozturk. Conflict networks. Journal of Public Economics, 126:104–113, 2015.
J. Fu, Q. Lu and Y. Pan. Team contests with multiple pairwise battles. American Economic
Review, 105(7):2120–2140, 2015.
M. Garfinkel and S. Skaperdas. Conflict without misperceptions or incomplete information:
How the future matters. The Journal of Conflict Resolution, 44(6):793–807, 2000.
M. Garfinkel and S. Skaperdas. The Oxford Handbook of the Economics of Peace and Conflict.
Oxford University Press, 2012.
E. Gibbon. The Decline and Fall of the Roman Empire. Strahan & Cadell, London, 1776.
C. Groh, B. Moldovanu, A. Sela, and U. Sunde. Optimal seedings in elimination tournaments.
Economic Theory, 49(1):59–80, 2012.
T. Hiller. Friends and enemies: a model of signed network formation. Theoretcial Economics,
12(3):1057–1087, 2017.
J. Hirshleifer. Conflict and rent-seeking success functions: Ratio vs. difference models of
relative success. Public Choice, 63(2):101–112, 1989.
J. Hirshleifer. Anarchy and its breakdown. Journal of Political Economy, 103:26–52, 1995.
T. Hobbes. Leviathan. 1651. Critical edition by Noel Malcolm in three volumes: 1. Editorial
Introduction; 2 and 3. The English and Latin Texts, Oxford University Press, 2012.
M. Howard. War in European History. Oxford University Press, Oxford, 2009.
D. Hume. Essays, Moral, Political and Literary. Cosimo, New York, 2006.
K. Huremovic. A noncooperative model of contest network formation. AMSE Working Papers
1521, Aix-Marseille School of Economics, Marseille, France, 2015.
M. Jackson and M. Morelli. Political bias and war. American Economic Review, 97(4):
1353–1373, 2007.
30
M. Jackson and S. Nei. Networks of military alliances, wars, and international trade. Pro-
ceedings of the National Academy of Sciences of the United States of America, 112(50):
15277–15284, 2015.
J. Jordan. Pillage and property. Journal of Economic Theory, 131:26–44, 2006.
I. Khaldun. The Muqaddimah: An Introduction to History. Bollingen Press, Princeton,
Princeton, New Jersey, 1989.
H. Kissinger. World Order. Penguin Books, London, 2015.
M. Konig, M. Rohner, D. Thoenig, and F. Zilibotti. Networks in conflict: Theory and evidence
from the great war of africa. Econometrica, 85(4):1093–1132, 2017.
K. Konrad. Strategy and Dynamic in Contests. London School of Economics Perspectives in
Economic Analysis. Oxford University Press, Canada, 2009.
K. Konrad and D. Kovenock. Multi-battle contests. Games and Economic Behavior, 66(1):
256–274, 2009.
D. Kovenock and B. Roberson. Conflicts with multiple battlefields. In M. Garfinkel and
S. Skaperdas, editors, Oxford Handbook of the Economics of Peace and Conflict, Oxford
Handbooks in Economics. Oxford University Press, New York, USA, 2012.
C. Krainin and T. Wiseman. War and stability in dynamic international systems. The Journal
of Politics, 78:1139–1152, 2016.
D. Levine and S. Modica. Conflict, evolution, hegemony and the power of the state. Working
Paper 19221, NBER, 2013.
D. Levine and S. Modica. Dynamics in stochastic evolutionary models. Theoretical Economics,
11:89–131, 2016.
M. E. Lewis. The Early Chinese Empires. Harvard University Press, Cambridge, 2010.
N. Machiavelli. The Prince. Everyman, New York, 1992.
P. Martin, T. Mayer, and M. Thoenig. Make trade not war? Review of Economics Studies,
75:865–900, 2008.
31
M. McBride and S. Skaperdas. Conflict, settlement, and the shadow of the future. Journal of
Economic Behavior & Organization, 105:75–89, 2014.
I. Morris and W. Scheidel. The dynamics of ancient empires: State power from Assyria to
Byzantium. Oxford University Press, Oxford, 2009.
N. Novta. Ethnic diversity and the spread of civil war. Journal of the European Economic
Association, 1074-1100, 2016.
W. Olszewski and R. Siegel. Large contests. Econometrica, 84(2):835–854, 2016.
M. Piccione and A. Rubinstein. Equilibrium in the jungle. Economic Journal, 117:883–896,
2007.
Polybius. The Histories: Volumes I-III. Loeb Classical Library. Harvard University Press,
Cambridge, Mass, 2010.
M. Shubik. Does the fittest necesasrily survive. In M. Shubik, editor, Readings in Game
Theory and Political Behavior, pages 43–46. Doubleday, 1954.
S. Skaperdas. Contest success functions. Economic Theory, 7:283–290, 1996.
Tacitus. Annals and Histories. Everyman’s Library, New York, 2009.
R. Thapar. Asoka and the Decline of the Mauryas. Oxford University Press, New Delhi, 1997.
R. Thapar. Early India: From the Origins to 1300. Penguin, London, 2002.
P. Turchin. War and Peace and War: The Rise and Fall of Empires. Penguin, New York,
2007.
S. Tzu. The Art of War. Penguin Classics, London, 2008.
32
Appendix: Proofs
Equilibrium existence and payoff uniqueness
Proof of Proposition 1. We start with introducing a natural partial order of precedence on the
set of ownership configurations, O, and on the set of states, O×2N . Given any two ownership
configurations, o ∈ O and o′ ∈ O, o v o
′ if and only if for all v ∈ V , either o(v) = o′(v)
or o(v) 6= o′(u), for all u ∈ V . Informally, if o and o
′ are ownership configurations such
that o′ is obtained from o by some rulers expanding their territories, then o v o′. Given
any two states, (o, P ) ∈ O× 2N and (o′, P ′) ∈ O× 2N , (o, P ) � (o′, P ′) if and only if either
o v o′ or o = o
′ and P ⊆ P ′. Informally, if state (o, P ) precedes state (o′, P ′) in the course
of the game, then (o, P ) � (o′, P ′). We will also use @ and ≺ to denote the strict orders
associated with the respective partial orders, defined above. Given an ownership configuration,
o ∈ O, let Succ(o) = {o′ ∈ O : o @ o′)} be the set of all ownership configurations that o
precedes. Let Succ(o) = Succ(o) ∪ {o}. Similarly, given a state (o, P ) ∈ O × 2N , let
Succ(o, P ) = {(o′, P ′) ∈ O × 2N : (o, P ) ≺ (o′, P ′)} be the set of all states that (o, P )
precedes, and let Succ(o, P ) = Succ(own, P ) ∪ {(own, P )}.Since O and O×2N are finite, there exist maximal elements of v and �. Take any strategy
profile, s, defined recursively on O×2N starting from the maximal elements of � as follows. If
(o, P ) is such that o is maximal according to v (i.e. there is only one active ruler at o) then,
for all i ∈ N , si(o, P ) = ε (the unique feasible sequence of i at o). Otherwise, let si(o, P ) be
any sequence that maximises i’s expected payoff given the continuation payoff determined by
s defined on the states in Succ(o, P ). Clearly any such strategy profile is well defined and is a
Markov perfect equilibrium of the game. Moreover, given the Markov perfection requirement
and since at each state there are no simultaneous moves (only one player is picked to make a
choice), every Markov equilibrium is a strategy profile of the form defined above.
We now turn to showing payoff equivalence of equilibria. Take any two Markov perfect
equilibria of the game, s and s′, and suppose that they are not payoff equivalent. Let (o, P ) ∈O × 2N be a maximal state, according to �, such that there exists a ruler i ∈ N \ P with
Πi(s | o, P ) 6= Πi(s′ | o, P ). Suppose that Πi(s | o, P ) > Πi(s
′ | o, P ) (the arguments for the
inverse inequality are symmetric and omitted). Then i could strictly improve his payoff under
s′ by choosing a strategy s′′i different to s′i at state (o, P ) only: s′′i (o, P ) = si(o, P ). Since
(o, P ) is a maximal state, according to �, for which Πi(s | o, P ) 6= Πi(s′ | o, P ), so for all
states in Succ(o, P ), s and s′ yield the same payoff to i and the payoff to i at (o, P ) depends
33
on his resources at (o, P ) and on his payoff at these states only. Thus Πi((s′−i, s
′′i ) | o, P ) =
Πi(s | o, P ) > Πi(s′ | o, P ), a contradiction with the assumption that s′ is a Markov perfect
equilibrium of the game. Hence for all i ∈ N and (o, P ) ∈ O× 2N\{i}, s and s′ must yield the
same payoff to i.
Incentives to fight
Proof of Proposition 2. We start with the rich rewarding case. The proof for the poor reward-
ing case is similar and omitted. Let p be a contest success function satisfying (5). Suppose
that p is rich rewarding and take any x, y ∈ R++ such that x > y. Rewriting (x+y)p(x, y) > x,
it is equivalent to 1
1+f(y)f(x)
> 11+ y
x. Further, this is equivalent to f(x)/x > f(y)/y. Hence rich
rewarding property is equivalent to f(x)/x being strictly on R++, that is to f exhibiting
increasing returns to scale.
We next turn to the timing results and consider the no-waiting case. The proof for the waiting
case is similar and omitted. Let p be a contest success function satisfying (5). Suppose that
p has the no-waiting property. Then, for any x, y, z ∈ R++,
f(x)
f(x) + f(y)
f(x+ y)
f(x+ y) + f(z)>
f(x)
f(x) + f(y + z).
In particular, the inequality holds for z = x so
f(x)
f(x) + f(y)
f(x+ y)
f(x+ y) + f(x)>
f(x)
f(x) + f(x+ y).
holds for any x, y ∈ R++. Dividing both sides by f(x) and multiplying them by the denomi-
nators we get
f(x+ y)(f(x) + f(x+ y)) > (f(x) + f(y))(f(x+ y) + f(x)).
Dividing both sides by f(x) + f(x+ y) yields
f(x+ y) > f(x) + f(y).
Thus f is super-additive.
34
Next suppose that f is super-additive. Then, for any y, z ∈ R++,
f(y + z) > f(y) + f(z).
Multiplying both sides of the inequality above by f(x+ y), for any x, y, z ∈ R++,
f(x+ y)f(y + z) > f(x+ y)(f(y) + f(z)).
Moreover,
f(x+ y)f(y + z) > f(x+ y)f(y) + f(x+ y)f(z) > f(x+ y)f(y) + (f(x) + f(y))f(z).
Adding f(x)f(x+ y) to both sides we get
f(x+ y) (f(x) + f(y + z)) > (f(x) + f(y)) (f(x+ y) + f(z)) .
This can be rewritten as
1
f(x) + f(y)
f(x+ y)
f(x+ y) + f(z)>
1
f(x) + f(y + z).
Multiplying both sides by f(x) we get
f(x)
f(x) + f(y)
f(x+ y)
f(x+ y) + f(z)>
f(x)
f(x) + f(y + z).
To complete the argument, we show that increasing returns to scale imply super-additivity
and that decreasing returns to scale imply sub-additivity. Suppose that f has increasing
returns to scale. So f(x)/x is strictly increasing on R++. For any x, y ∈ R++,
xf(x+ y) > (x+ y)f(x) and yf(x+ y) > (x+ y)f(y).
Adding the two inequalities and dividing both sides by x+ y we get f(x+ y) > f(x) + f(y),
that is f is strictly super-additive. The arguments for decreasing returns are similar and
omitted.
By what was shown above, rich rewarding property of p implies that f exhibits increasing
returns to scale which, in turn, implies that f is super-additive and, further, that p has
35
the no-waiting property. By similar argument, poor rewarding property implies the waiting
property.
Finally we turn to the order of attack result. We provide the proof for the poor-first case.
The arguments for the rich-first case are similar and omitted. Let x, y, z ∈ R++ with y > z
and suppose that p(x, y)p(x+ y, z) > p(x, z)p(x+ z, y). This may be rewritten as:
f(x)
f(x) + f(y)
f(x+ y)
f(x+ y) + f(z)>
f(x)
f(x) + f(z)
f(x+ z)
f(x+ z) + f(y)
Dividing both sides by f(x) and multiplying them by the denominators, we get
f(x+ y)(f(x) + f(z))(f(x+ z) + f(y)) > f(x+ z)(f(x) + f(y))(f(x+ y) + f(z))
This is equivalent to
f(y)f(x+ y)(f(x) + f(z)) + f(x)f(x+ z)f(x+ y) + f(z)f(x+ z)f(x+ y) >
f(z)f(x+ z)(f(x) + f(y)) + f(x)f(x+ y)f(x+ z) + f(y)f(x+ y)f(x+ z)
Subtracting f(x)f(x+ z)f(x+ y) + f(z)f(x+ z)f(x+ y) + f(y)f(x+ y)f(x+ z) from both
sides this is equivalent to
f(y)f(x+ y)(f(x) + f(z))− f(y)f(x+ y)f(x+ z) >
f(z)f(x+ z)(f(x) + f(y))− f(z)f(x+ z)f(x+ y)
Reorganizing and multiplying both sides by −1, this is equivalent to
f(z)f(x+ z)(f(x+ y)− (f(x) + f(y))) > f(y)f(x+ y)(f(x+ z)− (f(x) + f(z))).
Dividing both sides by (f(x+ y)− (f(x) + f(y)))(f(x+ z)− (f(x) + f(z))), this is equivalent
tof(z)f(x+ z)
f(x+ z)− f(x)− f(z)>
f(y)f(x+ y)
f(x+ y)− f(x)− f(y).
This completes the proof.
The timing part of Proposition 2 can be generalized to arbitrary sequences of fights: with
poor-first property attacking opponents in increasing order with respect to their resources is
36
optimal, while with rich-first property attacking them in the reversed order is optimal. This
is stated in the corollary below.
Corollary 2. Let m ≥ 3 and x0, x1, . . . , xm ∈ R++, be such that x1 < . . . < xm. Then, for
any permutation π : {1, . . . ,m} → {1, . . . ,m},
pseq(x0, xπ(1), . . . , xπ(m)
)≤
{pseq(x0, x1, . . . , xm), if p has poor-first property,
pseq(x0, xm, . . . , x1), if p has rich-first property.
with equality only if the permutations on both sides are the same.
Proof. We provide the proof for the poor-first property. The proof for the rich-first property is
similar and omitted. Assume p has poor-first property. Let π : {1, . . . ,m} → {1, . . . ,m} be a
permutation of {1, . . . ,m}. A pair of indices (i, j) ∈ {1, . . . ,m} such that i < j and π(i) > π(j)
is called an inverse of π. We will show that for any permutation π of {1, . . . ,m} with at least
one inverse there exists a permutation π′ of {1, . . . ,m} with less inverses that yields higher
pseq: pseq(x0, xπ(1), . . . , xπ(m)) < pseq(x0, xπ′(1), . . . , xπ′(m)). Since the identity is the unique
permutation of {1, . . . ,m} with no inverses, this implies the proposition. Throughout the
proof, given a permutation π and j ∈ {1, . . . ,m} we will use Xπ(j) to denote∑j
l=1 xπ(l).
So take any permutation π on {1, . . . ,m} with at least one inverse, (i, j). Then there exists
i ≤ k < j such that (k, k + 1) is also an inverse of π. Let π′ be a permutation of {1, . . . ,m}obtained from π by exchanging π(k) and π(k + 1), i.e. π′(k) = π(k + 1), π′(k + 1) = π(k),
and π′(l) = π(l) for l ∈ {1, . . . ,m} \ {k, k + 1}. There is at least one inverse less in π′ than
in π. Moreover, pseq(x0, xπ′(1), . . . , xπ′(m)
)= pseq
(x0, xπ′(1), . . . , xπ′(k−1)
)· p(Xπ′(k−1), xπ′(k)
)·
p(Xπ′(k), xπ′(k+1)
)·pseq
(Xπ′(k+1), xπ′(k+2), . . . , xπ′(m)
)= pseq
(x0, xπ(1), . . . , xπ(k−1)
)·p(xπ′(k−1), xπ′(k)
)·
p(Xπ′(k), xπ′(k+1)
)·pseq
(xπ(k+1), xπ(k+2), . . . , xπ(m)
). By poor-first property, this is greater than
pseq(x0, xπ(1), . . . , xπ(k−1)
)·p(xπ(k−1), xπ(k)
)·p(Xπ(k), xπ(k+1)
)·pseq
(Xπ(k+1), xπ(k+2), . . . , xπ(m)
)=
pseq(x0, xπ(1), . . . , xπ(m)
). This completes the proof.
Conquest and empire
We start by noting that the waiting and no-waiting properties extend to sequences of arbitrary
length. Formally, let m ≥ 3, x1, . . . , xm ∈ R++, and 1 ≤ i < j ≤ m such that i 6= 1 or j 6= m.
37
If p has the no-waiting property, then
pseq(x1, . . . , xi−1, xi, . . . , xj, xj+1, . . . , xm) > pseq
(x1, . . . , xi−1,
j∑l=i
xl, xj+1, . . . , xm
)(8)
Proof of Theorem 1. The proof proceeds in three steps.
Step 1: Fix some state o with |Act(o)| ≥ 2. For a strong ruler i, the optimal full attacking
sequence maximizes his payoffs across all attacking sequences. Moreover, in generic case, it is
a unique maximizer.
Let o be a state with |Act(o)| = m ≥ 2. Take an active ruler j0 ∈ Act(o) with maximal
amount of resources Rj0(o). For generic resource values, such a ruler is unique. Pick a full
attacking sequence j1, . . . , jm−1 consisting of rulers in Act(o) \ {j0} that is feasible for j0 in
G under o (clearly such a sequence exists because G is connected). Since j0 has maximal
amount of resources so, for all 1 ≤ k ≤ m− 1, we have
k−1∑l=0
Rjl(o) ≥ Rjk(o). (9)
The expected payoff to ruler j0 from the attacking sequence is
πj0(o | j1, . . . , jm−1) =
(m−1∑l=0
Rjl(o)
)m−1∏k=1
p
(k−1∑l=0
Rjl(o), Rjk(o)
)
= Rj0(o)m−1∏k=1
p
(k−1∑l=0
Rjl(o), Rjk(o)
)(∑kl=0Rjl(o)∑k−1l=0 Rjl(o)
). (10)
Since p is rich rewarding, so
p
(k−1∑l=0
Rjl(o), Rjk(o)
)(∑kl=0Rjl(o)∑k−1l=0 Rjl(o)
)≥ 1, (11)
with equality only if k = 1 and Rj0(o) = Rj1(o).
At every step in the sequence, the expected resources are growing. So, for generic resource
values, there is a full attacking sequence that dominates any partial attacking sequence. By
definition, the optimal full attacking sequence maximizes payoffs across all attack sequences.
The first step has a powerful implication: in any state with 2 or more active rulers there
is at least one ruler who has a strict incentive to attack, given that other rulers do not attack.
38
Hence, in equilibrium, there must exist a hegemon.
In the dynamic game, in principle, a strong ruler may prefer to wait and allow others to
move and then attack later. The next step shows that an optimal full attacking sequence
dominates all such waiting strategies.
Step 2: Fix some state o with |Act(o)| ≥ 2 and a set or rulers, P . For any ruler i ∈ N \ Pstrong at o, an optimal full attacking sequence is a dominant choice at (o, P ). Moreover, the
choice is strictly dominant if |Act(o)| ≥ 3.
Fix some state o. Let σi(o) be the optimal sequence of ruler i at o, assuming that the
game ends after i executes the sequence (successfully or not). In other words, σi(o) is the
myopic optimal sequence of ruler i at o. Notice that this sequence is independent of the set of
rulers who chose peace prior to i’s move at a round at the state o. Let πi(o) = πi(o | σi(o))
denote the optimal myopic payoff ruler i can attain at o.
Claim. The optimal myopic payoff is the highest that ruler i can hope to attain, i.e., for any
state, o, and any set of rulers, P ⊆ N \ {i}, πi(o) ≥ Πi(s | o, P ) for any feasible strategy
profile s. Moreover, if i is strong and there are at least three active rulers, then the inequality
is strict.
The proof is by induction on the number of active rulers. For the induction basis, we show
that the claim holds for 2 active rulers. If i is the richer ruler then, from the rich rewarding
property, his myopic optimal strategy is to attack. It is also clear that attacking yields strictly
higher payoffs if the other ruler does not attack, and weakly higher payoffs if the other ruler
does attack. If i is the poorer ruler then not attacking is the optimal myopic strategy. In case
the richer ruler attacks, the expected payoff to i is less due to the rich rewarding property.
That completes the argument for 2 active rulers.
For the induction step, suppose that the claim holds for all y ≤ X, where X ≥ 2, active
rulers: we will show that it also holds for X + 1 active rulers. Given state o′, set of rulers,
P ′, and strategy profile, s′, we will use Atck(s′,o′, P ′) to denote the set of rulers choosing
attack at (o′, P ′) under s′, i.e. Atck(s′,o′, P ′) = {j ∈ N \ P ′ : s′j(o′, P ′) 6= ε}.23 Fix some
state o with X + 1 active rulers and a set of rulers, P such that Act(o) \ P 6= ∅. Take an
active ruler i ∈ Act(o) \ P and any strategy profile s. If for all P ′ ⊆ N such that P ⊆ P ′,
Atck(s,o, P ′) = ∅, i.e. all players choose peace following P at o, then the claim follows,
23Throughout the proofs we use the standard notation, ε, to denote empty sequences.
39
because σi(o) is at least as good as the empty sequence at o:
πi(o) ≥ πi (o | σi(o)) ≥ πi (o | ε) = Πi(s | o, P ). (12)
Moreover, by Step 1, the inequality is strict if i is strong.
For the remaining part of the argument assume that there exists P ′ ⊆ N with P ⊆ P ′ such
that Atck(s,o, P ′) 6= ∅. We will establish that πi(o) ≥ Πi(s | o, P ). Given a set of rulers P ′
and ruler j0 ∈ N \ P ′ such that sj0(o, P′) 6= ε , let Πi(s | o, P ′, j0) denote the expected payoff
to ruler i from strategy profile s conditional on ruler j0 being selected to move at (o, P ′) and
q(j0, P′ | o, s) denote the probability that j0 is picked after P ′ at o under s. Then
Πi(s | o, P ) =∑
P ′⊆N s.t. P⊆P ′
∑j0∈Atck(s,o,P ′)
q(j0, P′ | o, s)Πi (s | o, P ′, j0) +1−
∑P ′⊆N s.t. P⊆P ′
∑j0∈Atck(s,o,P ′)
q(j0, P′ | o, s)
πi (o | ε) (13)
As we established above, πi(o) ≥ πi(o | ε), with strict inequality if i is strong. Thus to show
the claim, it is enough to show that
πi(o) ≥ Πi (s | o, P ′, j0) , (14)
for each P ′ ⊆ N with P ⊆ P ′ and each attacking ruler j0 ∈ Atck(s,o, P ′), with strict
inequality for at least one P ′ and j0 ∈ Atck(s,o, P ′) in the case of i being strong.
So take any set of rulers, P ′ ⊆ N with P ⊆ P ′ and any ruler j0 ∈ Atck(s,o, P ′). Three
cases are possible:
(i). j0 6= i and i is not in the attacking sequence sj0(o) of j0,
(ii). j0 6= i and i is in the attacking sequence sj0(o) of j0,
(iii). j0 = i.
Case (i). Ruler j0 is different to i and does not have i in his attacking sequence sj0(o). Let
F (o′ | s,o, P ′, j0) be the probability of reaching ownership state o′ in the next round from
state o under strategy profile s when j0 is selected to move after P ′ (and executes attacking
40
sequence sj0(o, P′)). Then
Πi (s | o, P ′, j0) =∑o′∈O
F (o′ | s,o, P ′, j0) Πi(s | o′). (15)
To show (14) it is enough to show that
πi(o) ≥ Πi(s | o′) = Πi(s | o′,∅), (16)
for each state o′ that can be reached in the next round with positive probability from o when
j0 plays the attacking sequence sj0(o, P′) after P ′ at o. We will show that the inequality is
strict when i is strong.
Ownership state o′ is reached after at least one fight and so has at most X active rulers.
Hence, by the induction hypothesis, πi(o′) ≥ Πi(s | o′,∅), and so to show (16) it is enough
to show that
πi(o) ≥ πi(o′). (17)
Take an optimal myopic sequence, σi(o′), of i at o′. There are two sub-cases to be considered.
(a) Sequence σi(o′) does not contain the rulers in the sequence of fights that leads to o
′.
This means, in particular, that σi(o′) is not a full attacking sequence. Hence, by Step 1, i is
not strong.
Since σi(o′) does not contain the rulers in the sequence of fights that leads to o′, it can be
executed at state o. By optimality of σi(o) at o
πi(o) = πi (o | σi(o)) ≥ πi (o | σi(o′)) = πi (o′ | σi(o′)) = πi(o
′). (18)
(b) Sequence σi(o′) contains at least one ruler in the sequence of fights that leads to o
′.
This is true, in particular, when i is strong because, by Step 1, σi(o′) must be a full attacking
sequence then.
Since σi(o′) contains at least one ruler in the sequence of fights that leads to o′, so σi(o
′) =
σ1i (o′), k, σ2
i (o′), where k is the ruler who won the sequence of fights leading to o
′. We can
construct a sequence σ′ = σ1i τσ
2i that is feasible for i at o, with τ being a sequence of rulers
involved in the sequence of fights leading to o′. By point 1 of Proposition 2 p has the no-
waiting property. As we observed prior to the proof of the theorem, the no-waiting property
extends to sequences of fights of arbitrary length – (8). Given this observation, σ′ yields a
strictly higher payoff than σi(o′). By construction, σi(o) is an optimal myopic strategy for i
41
at o and so payoff dominates σ′ at o. Hence
πi(o) = πi (o | σi(o)) ≥ πi(o | σ′) > πi (o′ | σi(o′)) = πi(o
′). (19)
Hence (17) and, consequently, (16) hold with strict inequality.
Case (ii). Ruler j0 is different to i and has i in his attacking sequence sj0(o). Let sj0(o) =
j1, . . . , jm be the sequence selected by j0 at o under strategy sj0 . Then i = jk for some
1 ≤ k ≤ m. Given l ∈ {1, . . . ,m}, let ol be the state reached after j0 looses the l’th fight in
the sequence. The expected payoff to i from s at o given that j0 is selected to move after set
P ′ or rulers is equal to
Πi (s | o, P ′, j0) =k−1∑l=1
F(ol | s,o, P ′, j0
)Πi
(s | ol
)+(
1−k−1∑l=1
F(ol | s,o, P ′, j0
))p
(ri(o),
k−1∑l=0
rjl(o)
)Πi
(s | ok
), (20)
where j1, . . . , jk−1 are the rulers attacked by j0 prior to attacking i.
Hence to show (14) it is enough to show that (16) holds for all o′ = ol, l ∈ {1, . . . , k− 1},
reachable after a sequence of fights of j0 in which j0 looses before facing i, and that
πi(o) ≥ p
(ri(o),
k−1∑l=0
rjl(o)
)Πi
(s | ok
). (21)
holds for ok, reachable by a sequence of fights of j0 in which i is attacked by j0 and wins. (16)
is shown by the same arguments as in point (ii) above. In particular, the inequality in (16)
is strict when i is strong. For (21), let τ be a sequence of rulers {j0, . . . , jk−1} feasible to i
at o (clearly such a sequence exists). Then sequence σ′ = τσi(ok), consisting of τ and an
optimal myopic sequence of i at ok, is feasible for i at o. By the no-waiting property and its
generalization, (8), τ yields at least the same payoff to i as the sequence of fights that leads
to o′ (the inequality is strict, unless k = 1). Combining this with the induction hypothesis we
42
get
πi(o) ≥ πi(o | τσi(ok)) ≥ p
(Ri(o),
k−1∑l=0
Rjl(o)
)πi(ok | σi
(ok))
≥ p
(Ri(o),
k−1∑l=0
Rjl(o)
)Πi
(s | ok
), (22)
with strict inequality, unless k = 1.
Case (iii). Ruler i is picked to move at o after P ′. The strategy chosen by i under strategy
profile s at (o, P ′) is si(o, P′). Let o′ be the state that is reached if i wins all the attacks in
sequence si(o, P′). Then sequence σ′ = si(o, P
′)σi(o′), consisting of si(o, P
′) and an optimal
myopic sequence of i at o′, is feasible for i at o. State o′ is reached after at least one fight and
has at most X active rulers. By the induction hypothesis, πi(o′) ≥ Πi(s | o′) and it follows
that
πi(o) ≥ πi (o | si(o)σi(o′)) ≥ πi (o
′ | σi(o′)) = πi(o′) ≥ Πi(s | o′). (23)
The inequality is strict unless the sequence si(o)σi(o′) is the same as the optimal myopic
sequence of i at o.
To complete the proof of the claim, we argue that πi(o) > Πi(s | o, P ) if i is strong and
there are at least 3 active rulers at o. As we established above, if i is strong then (14) holds
with equality in two cases only: j0 = i and si(o, P ) is the optimal myopic sequence of i at o,
or j0 = j 6= i, j0 attacks i first under sj0(o, P ) and j0 is the first ruler to be attacked by i
under his optimal myopic sequence of attacks. Generically the second case is possible for at
most one ruler other then i. Hence with at least three active rulers there is at least one for
which the inequality in (14) is strict. This completes the proof of the claim.
From Step 1, we know that in any state o, there exists a strong ruler for whom the full
attacking sequence is the optimal stand alone strategy and it is optimal for him to choose it
after any set of rulers P at o. It now follows from the claim above that for this strong ruler
the optimal full attacking sequence dominates all other strategies, and the domination is strict
if there are at least three active rulers at o. The final step in the proof takes up non-strong
rulers. We show that faced with rulers such that at every state at least one of them attacks,
every ruler will find it profitable to choose an optimal full attacking sequence.
Step 3: Let i ∈ N be a ruler, s be a strategy profile such that for every state o and
for every permutation of N , j1, . . . , jn, there exists k ∈ {1, . . . , n} such that jk 6= i and
43
sjk(o, {j1, . . . , jk−1}) 6= ε. Let si be a best response of i to s−i. Then for every state o
such that i ∈ Act(o) and |Act(o)| ≥ 3, and for every set of rulers, P ⊆ N \ {i} such that
Atck(s,o, P ) \ {i} 6= ∅, si(o, P ) is an optimal full attacking sequence of i at o.
Let i ∈ N be a ruler and let s−i be a strategy profile of the other rulers, as stated above.
The assumption means that at every state o, for any draw of rulers, with probability 1 a ruler
other than i would choose attack if i would not. Let si be a strategy such that at every state
o where ruler i is active and there are at least three active rulers, and for every P ⊆ N \ {i}with Atck(s,o, P ) \ {i} 6= ∅, si(o, P ) is an optimal full attacking sequence for i. We show
that for any other strategy, s′i, of ruler i, every state o ∈ O with |Act(o)| ≥ 2, and every set
of rulers P ⊆ N \ {i} such that Atck(s,o, P ) \ {i} 6= ∅,
Πi ((si, s−i) | o, P ) ≥ Πi((s′i, s−i) | o, P )), (24)
with strict inequality when |Act(o)| ≥ 3. Notice that, by Step 2, the claim holds if i is strong
at o. For the remaining part of the proof we will consider rulers who are not strong at the
given states.
The argument is by induction on the number of active rulers. As it proceeds along lines
similar to Step 2, it is omitted.
Proof of Proposition 3. A sequence σ ∈ R∗ is strong if either σ = ε or σ = x0, . . . , xm and for
all k ∈ {1, . . . ,m},∑k−1
j=0 xj > xk. A sequence σ ∈ R∗ is weak if it is not strong.
Let p(x, y | γ) = xγ
xγ+yγ. Since
∂p
∂γ=
(xγyγ
xγ + yγ
)(ln(x)− ln(y))
and
limγ→+∞
xγ
xγ + yγ= lim
γ→+∞
1
1 +(yx
)γ =
{1, if x > y
0, if x < y.
so for x > y, p(x, y | γ) is increasing and converges to 1 when γ → +∞, and for x < y, p(x, y |γ) is decreasing and converges to 0 when γ → +∞. In addition, for any strong sequence σ,
pseq(σ | γ) is increasing when γ is increasing. This is because for all k ∈ {1, . . . ,m},∑k−1
j=0 xj >
xk, and so limγ→+∞∏m
k=1 p(∑k−1
j=0 xj, xk
∣∣∣ γ) = 1 and∏m
k=1 p(∑k−1
j=0 xj, xk
∣∣∣ γ) is increasing
when γ is increasing. On the other hand, for any weak σ = x0, . . . , xm, limγ→+∞ pseq(σ | γ) = 0.
44
This is because there exists k ∈ {1, . . . ,m} such that∑k−1
j=0 xj < xk and for any such k,
limγ→+∞ p(∑k−1
j=0 xj, xk
∣∣∣ γ) = 0. Since for all other k ∈ {1, . . . ,m}, p(∑k−1
j=0 xj, xk
∣∣∣ γ) ≤ 1
so limγ→+∞∏m
k=1 p(∑k−1
j=0 xj, xk
∣∣∣ γ) = 0. Consequently, for any non-empty sequence σ =
x0, . . . , xm,
limγ→+∞
pseq(σ | γ) =
{1, if σ is strong
0, if σ is weak.
The claim on probability of hegemony for strong rulers now follows.
Proof of Proposition 4. Part 1: The argument presented here is true for general contest
functions. Take any ownership configuration, o ∈ O, and any active ruler, i ∈ Act(o). Given
a state (o, P ) ∈ O × 2N\{i}, an attacking sequence, σ, is an optimal attacking sequence if
it maximises the payoff of i at (o, P ) across all attacking sequences that are feasible to i at
o and given the continuation payoffs determined by s on the states in Succ(o, P ). Notice
that if a sequence is an optimal attacking sequence for i at (o, P ), then it is an optimal
attacking sequence of i at (o, P ′), for any P ′ ⊆ N . Thus its optimality depends on the
ownership configuration and the expected payoff determined by s on ownership configurations
o′ ∈ Succ(o), only. Clearly, at every state (o, P ) ∈ O× 2N\{i} an expected payoff maximising
ruler chooses between the empty sequence (peace) and an optimal attacking sequence at o.
Given ownership configuration o, let E(s,o) be the set of rulers, active at o, for whom an
optimal attacking sequence at o yields higher payoff than the empty sequence. It is easy to see
that if E(s,o) = ∅ and s is an equilibrium, then si(o, P ) = ε, for all i ∈ N and P ∈ 2N\{i}.
On the other hand, suppose that E(s,o) 6= ∅ and take any sequence i1, . . . , in of rulers from
N . Let ik be the last ruler from E(s,o) in the sequence. Generically, no ruler is indifferent
between peace and an optimal attacking sequence. Hence, if s is an equilibrium then, for
every l > k, sil({i1, . . . , il−1}) is the empty sequence and, consequently, sik({i1, . . . , ik−1}) is
an optimal attacking sequence of ik at o under the continuation of s. Hence if E(s,o) 6= ∅then o is conflictual under s.
Part 2: Let p be a poor rewarding contest success function satisfying (5). Then p(x, y) =
f(x)/(f(x) + f(y)) and, by Propoistion 2, f(x)/x is decreasing. Since f(x)/x is decreasing
and positive on R++ so limx→+∞ f(x)/x exists and is finite. Let limx→+∞ f(x)/x = L.
Consider a sequence of fights where a ruler with x ∈ R++ resources first fights a ruler with
y ∈ R++ resources and then fights with m ≥ 1 rulers with resources z1, . . . , zm ∈ R++. The
45
expected payoff to the rulers with x resources from such a sequence of fights is is equal to
π(x, y, z1, . . . , zm) = pseq(x, y, z1, . . . , zm)(x+ y + z1 + . . .+ zm)
= x · f(x)
f(x) + f(y)· x+ y
x·m∏i=1
f(x+ y +
∑i−1j=1 zj
)f(x+ y +
∑i−1j=1 zj
)+ f(zi)
·x+ y +
∑ij=1 zj
x+ y +∑i−1
j=1 zj
.
We will show that for sufficiently large y, π(x, y, z1, . . . , zm) > x. We consider two cases
separately: L > 0 and L = 0.
Suppose first that L > 0. Notice that
limy→+∞
f(x)
f(x) + f(y)
x+ y
x= lim
y→+∞
f(x)x
f(x)y
+ f(y)y
(x
y+ 1
)=
f(x)x
L> 1.
Similarly
limy→+∞
f(x+ y +
∑i−1j=1 zj
)f(x+ y +
∑i−1j=1 zj
)+ f(zi)
·x+ y +
∑ij=1 zj
x+ y +∑i−1
j=1 zj=L
L= 1.
Hence limy→+∞ π(x, y, z1, . . . , zm) = t > x and so for sufficiently large y, π(x, y, z1, . . . , zm) >
x.
Second, suppose that L = 0. After winning the conflict with the ruler with y resources,
in every subsequent conflict in the sequence the starting ruler has higher resources than his
opponent. Hence the probability of winning each of these conflicts is more than 1/2. In the
event of winning all the conflicts in the sequence, the starting ruler owns at least x+y+∑m
j=1 zj
resources. By these observations π(x, y, z1, . . . , zm) ≥(
12m
) ( f(x)f(x)+f(y)
)(x + y). On the other
hand, since L = 0 so, for sufficiently large y,
f(y)
y+
(1− 1
2m
)f(x)
y<
1
2mf(x)
x.
Multiplying both sides by y/f(x) and reorganizing, this is equivalent to
f(y)
f(x)+ 1 <
1
2m
(1 +
y
x
).
Taking the inverses of both sides and then multiplying both sides by (x + y)/2m, this is
46
equivalent to (1
2m
)(f(x)
f(x) + f(y)
)(x+ y) > x.
Hence, for sufficiently large y, π(x, y, z1, . . . , zm) > x.
Now, let G be a connected network over the set of nodes, V , and let r ∈ R++ be a resource
endowment. Fix any vertex v ∈ V . Take any ownership configuration o ∈ O. If there is a
ruler who owns all the vertices under o then we are done. Assume otherwise. There are at
least two active rulers under o, |Act(o)| ≥ 2. Let i be the ruler owning vertex v, o(v) = i, and
let j ∈ Act(o) be any active neighbor of i under o. Let σ be a permutation of Act(o) \ {j}starting with i. Sequence σ is a full attacking sequence of j at o. By what we have shown
above, if rv is sufficiently large, then Π(j,o;σ) > Rj(o) and so by choosing σ ruler j strictly
increases his expected payoff. Since at every ownership configuration o with at least two
active rulers there exists a ruler who can increase his expected resources by choosing attack,
so every equilibrium outcome is hegemony.
Part 3: Let v ∈ V be a vertex and let G be a star network with centre v. Let p be a Tullock
contest success function with γ ∈ (0, 1). Take any y > 0. Let the resource vector r be such
that ru = y, for each spoke u ∈ V \ {v}, and rv = x, for the centre. We will show that there
exists (a range of values of) x such that there is an equilibrium where each ruler chooses peace
in the initial ownership configuration. Similarly, we will show that there exists (a range of
values of) x such that there is an equilibrium where each ruler at a spoke chooses a sequence
of fights that leads to a ownership configuration with peace (so we have war followed by peace
in equilibrium).
The expected payoff from a full attacking sequence of m fights to a ruler owning a spoke
in a star over at least m + 1 vertices, when each spoke is endowed with y resources and the
centre is endowed with x resources, is
ϕ(x, y,m) = (x+my)p(y, x)m−1∏i=1
p(x+ iy, y) = (x+my)
(yγ
xγ + yγ
)m−1∏i=1
((x+ iy)γ
(x+ iy)γ + yγ
)
The key to the constructions of resource endowments enabling equilibria described above is
the following claim:
Claim. For all m ≥ 2, γ ∈ [0, 1), and y > 0, there exists a unique x∗m = x∗m(y, γ) > y, such
47
that
ϕ(x, y,m)
< y if x ∈ (y, x∗m),
= y if x = x∗m,
> y if x > x∗m.
(25)
Moreover, x∗m+1(y, γ) > x∗m(y, γ).
Before proving the claim, we provide the construction of resource endowments. Taking
any x ∈ (max(y, x∗n−2 − y), x∗n−1) guarantees that no ruler has incentives to engage in a full
attacking sequence (and the interval is non-empty, as x∗n−2 > y, for n ≥ 4). Moreover, after at
least one fight, every ruler at a spoke has incentives to fight if no other ruler fights, as a full
attacking sequence yields him expected payoff higher than y. Thus any ruler deviating from
peaceful strategy profile leads to fight till hegemony, which is not profitable for the deviating
ruler. Therefore there is an equilibrium where all rulers choose peace in the initial ownership
configuration. Similarly, taking any x ∈ (max(0, x∗n−3−2y), x∗n−2−y) guarantees that after one
fight by a spoke, an ownership configuration with resources at the centre as described above
is reached. Moreover, at such a state, no ruler has incentives to engage in a full attacking
sequence. Thus there is an equilibrium where (1) in the initial state each ruler owning a spoke
chooses to attack the centre and the ruler owning the centre chooses peace, (2) in the state
with n − 1 vertices every vertex chooses peace, and (3) in any state with at most n − 2 at
least one vertex chooses attack. In this equilibrium there is one conflict followed by peace.
Notice that the two constructions given above are generic: analogous argument could be
conducted if spokes were endowed with resource sufficiently close to each other and the centre
was endowed with resources within a range close to the range given in the construction above.
We now provide the proof of the claim. To this end, we establish four properties of function
ϕ, from which the claim follows. Fix any γ ∈ [0, 1).
First, we show that, for all x, y ∈ R++ and m ≥ 3, ϕ(x, y,m) < ϕ(x, y,m−1). Notice that,
y1−γ ≤ (x+(m−1)y)1−γ. Multiplying both sides by yγ(x+(m−1)y)γ we get y(x+(m−1)y)γ <
yγ(x + (m − 1)y). Reorganizing, we obtain (x + my)(x + (m − 1)y)γ < ((x + (m − 1)y)γ +
yγ)(x+ (m− 1)y). Dividing both sides by the RHS we get(
x+myx+(m−1)y
)((x+(m−1)y)γ
(x+(m−1)y)γ+yγ
)< 1.
This, together with the fact that ϕ(x, y,m) = ϕ(x, y,m − 1)(
(x+(m−1)y)γ(x+(m−1)y)γ+yγ
)((x+my)
(x+(m−1)y)
)yields ϕ(x, y,m) < ϕ(x, y,m− 1).
Second, we show that ϕ is strictly increasing in x for x > y. First derivative of ϕ with
48
respect to x is
∂ϕ
∂x=
(γyγ
xγ + yγ
)(x+my)
m−1∏i=1
((x+ iy)γ
(x+ iy)γ + yγ
)((
1
γ(x+my)
)−(
xγ−1
xγ + yγ
)+
m−1∑j=1
yγ
(x+ jy)((x+ jy)γ + yγ)
). (26)
Since γ ∈ [0, 1) so (1− γ)(x + y) > 0. Reorganizing we get x + y + (m− 1)γy > γ(x + my).
Dividing both sides by γ(x+ y)(x+my) we get 1γ(x+my)
+ (m−1)y(x+y)(x+my)
> 1x+y
Since x > y and
γ ∈ [0, 1) so (x/y)1−γ > 1 and so 1x+y
> 1
x+y(xy )1−γ = xγ−1
xγ+yγ. Hence
1
γ(x+my)+
(m− 1)y
(x+ y)(x+my)>
xγ−1
xγ + yγ. (27)
Notice that
(m− 1)y
(x+ y)(x+my)=
(1
x+ y
)−(
1
x+my
)=
m−1∑i=1
(1
x+ iy
)−
m∑i=2
(1
x+ iy
)
=m−1∑i=1
((1
x+ iy
)−(
1
x+ (i+ 1)y
))=
m−1∑i=1
(y
(x+ iy)((x+ iy) + y)
)
Moreover, for γ ∈ [0, 1), x > y, and i ≥ 1,
y
(x+ iy)((x+ iy) + y)=
1
(x+ iy)((
xy
+ i)
+ 1) < 1
(x+ iy)((
xy
+ i)γ
+ 1)
=yγ
(x+ iy)((x+ iy)γ + yγ)
Thus(m− 1)y
(x+ y)(x+my)<
m−1∑i=1
(yγ
(x+ iy)((x+ iy)γ + yγ)
)which, together with (27), implies
1
γ(x+my)+
m−1∑i=1
(yγ
(x+ iy)((x+ iy)γ + yγ)
)>
xγ−1
xγ + yγ.
49
Therefore, by that and (26), ∂ϕ/∂x > 0 for all x > y and so ϕ is increasing in x on (y,+∞).
Third, we show that for all y ∈ R++ and m ≥ 3, limx→+∞ ϕ(x, y,m) = +∞. To see that no-
tice that limx→+∞∏m−1
i=1 p(x+iy, y) = 1 and limx→+∞ p(y, x)(x+my) =
(yγ
1+( yx)γ
)(x1−γ +m
(yxγ
))=
+∞, and so the property follows.
Fourth, we show that ϕ(y, y,m) < y. To see that we start with
ϕ(y, y,m) =
(1
2
)(m+ 1)y
m−1∏i=1
((i+ 1)γ
(i+ 1)γ + 1
)=
(1
2
)(m+ 1)y
m∏i=2
(iγ
iγ + 1
).
Since iγ
iγ+1= 1−
(1
iγ+1
), γ ∈ [0, 1), i ≥ 1, so iγ/(iγ + 1) is increasing in γ. Hence ϕ(y, y, n) <(
12
)(m+ 1)y
∏mi=2
(ii+1
)=(12
)y(n!n!
)2 = y.
By the four properties of ϕ, established above, for all m ≥ 2, γ ∈ [0, 1), and y > 0,
there exists a unique x∗m = x∗m(y, γ) > y, such that (25) holds. Moreover, since for all
x, y ∈ R++ and m ≥ 3, ϕ(x, y,m) < ϕ(x, y,m − 1), and since ϕ is increasing in x for x > y,
x∗m+1(y, γ) > x∗m(y, γ). This completes the proof.
Short attack sequences
Proof of Proposition 5. Throughout the proof we use the precedence relations on ownership
configurations and states, as well as the sets Succ and Succ introduced in proof of Proposi-
tion 1.
Notice that if i ∈ Act(o) is the unique strong ruler at o, then for all o′ ∈ Succ(o) there
is exactly one strong ruler in Act(o′) and if i ∈ Act(o′) then i is strong. This is because no
weak ruler has a strong full attacking sequence and therefore no such ruler can become strong,
unless he wins a conflict with a strong ruler (in which case he replaces the unique strong ruler
in the subsequent state).
Given a ruler i ∈ N and an ownership configuration, o, a strategy si of i is an attacking
strategy at o if, for every ownership configuration o′ ∈ Succ(o) such that i ∈ Act(o) and
|Act(o)| ≥ 2, and every set of rulers P ∈ 2N\{i}, si(o′, P ) 6= ε. Thus, at state o and at any
state following o in the course of the game, i never chooses to stay peaceful under si, unless
he is not active or is the unique active ruler.
Given a ruler i ∈ N , an ownership configuration, o, and a strategy profile of the other
ruler, s−i, we define an attacking strategy si that is a best attacking response of i to s−i at o.
The strategy is defined recursively on the set of states Succ(o,∅), starting from the maximal
50
elements under �. If (o′, P ) is such that o′ is maximal according to v in Succ(o) then, for
all i ∈ N , si(o′, P ) = ε (the unique feasible choice of i at o). Otherwise, let si(o, P ) be any
neighboring ruler attacking whom maximises i’s expected payoff across all neighbors of i at
o, given the continuation payoff determined by s = (si, s−i) defined on states in Succ(o, P ).
Notice that if j is such a ruler at (o, P ) then, for any P ∈ 2N\{i}, attacking j maximises i’s
expected payoff across all neighbors of i. Moreover, generically, such a neighbor is unique.
Now we are ready to give main part of the proof. First we show, for any strategy profile,
s, any ownership configuration, o ∈ O, any ruler i ∈ N , and any set of rulers P ⊆ N \ {i},that if i is the unique strong ruler at o then any best attacking response, s∗i , of i to s−i at o
yields i an expected payoff greater than Ri(o).
The proof is by induction on the number of active rulers at o. For the induction basis,
suppose that |Act(o)| = 2 and that i is the single strong ruler at o. Let j be the other active
ruler. Since p is rich rewarding and i is strong, the other active ruler is weak and attacking
him increases i’s payoff in expectation. Thus the claim holds.
For the induction step, take any 2 < m ≤ n suppose that the claim holds for any ownership
configuration o with |Act(o)| < m active rulers. Take any ownership configuration, o ∈ O,
with a unique strong ruler, i ∈ Act(o). Notice that since s∗i is an attacking strategy, so
s∗i (o, P ) 6= ε, for all P ∈ 2N\{i}. Hence, with probability 1, a ruler choosing attack will be
selected at o. Thus the strategy profile s = (s∗i , s) determines a probability distribution
Q(· | s,o) on the set A(o) = {(j, k) ∈ Act(o) × Act(o) : j 6= k} where, given (j, k) ∈ A(o),
Q(j, k | s,o) is the probability that ruler j attacks ruler k at o. Given two rulers, j, k ∈ Act(o),
active at o let o[j → k] denote the ownership configuration resulting from j wining a conflict
with k. The expected payoff to i at o, Πi(s | o), is equal to
Πi(s | o) =∑
(j,k)∈A(o)j 6=i,k 6=i
Q(j, k | s,o)(p(Rj(o), Rk(o))Πi(s | o[j → k]) +
p(Rk(o), Rj(o))Πi(s | o[k → j]))
+∑(j,i)∈A(o)
Q(j, i | s,o)p(Ri(o), Rj(o))Πi(s | o[i→ j]) +
Q(i, s∗i (o, P ) | s,o)p(Ri(o), Rs∗i (o,P )(o))Πi(s | o[i→ s∗i (o, P )])
By the observation at the beginning of the proof, i remains a unique strong ruler at each
51
ownership configuration o[j, k] with (j, k) ∈ A(o) such that j 6= i and k 6= i. Similarly, i
remains a unique strong ruler at each ownership configuration o[i, j] with j ∈ A(o). Thus, by
the induction hypothesis, for all (j, k) ∈ A(o), Πi(s | o[j → k]) > Ri(o[j → k]). In the case of
j 6= i and k 6= i, Ri(o[j → k]) = Ri(o). In the case of k = i, p(Ri(o), Rj(o))Πi(s | o[i→ j]) >
p(Ri(o), Rj(o))Ri(o[j → k]) = p(Ri(o), Rj(o))(Ri(o) + Rj(o)) > Ri(o), as Ri(o) > Rj(o)
and i is rich rewarding. Hence Πi(s | o) > Ri(o).
Now, suppose that there is a unique strong node in G under resource endowment r. Then
there is a unique strong ruler at the ownership configuration. Take any equilibrium s of the
game. By the observation above, there is a unique strong ruler at every ownership configura-
tion o ∈ O. In addition, point 1 of Proposition 4 extends immediately to the short sequence of
attack (the proof does not make any assumptions about the sequences that the rulers choose).
Hence in the short sequence model, like in the basic model, every ownership configuration is
either peaceful or conflictful under s. Take any peaceful ownership configuration o. It must
be that there is a unique active ruler at o as otherwise, by what was shown above, if no
other active ruler attacks his neighbor, the unique strong ruler attacks one of his neighbors.
Hence there is fight till hegemony under s. By generic uniqueness of equilibrium payoffs, the
probability of becoming a hegemon is generically unique.
Losses in war
Proof of Proposition 6. Given set of vertices U ⊆ V and resource endowment r let rU =∑v∈U rv. Also, given a resource endowment r over a set of vertices V let Z(r) = {(rU , rU ′) :
U,U ′ ∈ 2V \ {∅}, U ∩ U ′ = ∅}.Point 1: Fix the set of vertices V and resource endowment r. Assume first that p is rich
rewarding. Take any x, y ∈ R++. Since p(x, y)(x+ y) > x so δ1x,y = 1− x/((x+ y)p(x, y)) > 0
and for any δ ∈ (0, δ1x,y), p(x, y)(1 − δ)(x + y) > x. Next, assume that p has the no-waiting
property. Take any x, y, z ∈ R++. By monotonicity and continuity of p, p(x, y)p((1− δ)(x +
y), z)((1 − δ)(x + y) + z) is continuous and decreasing in δ. Moreover, by the no-waiting
property, p(x, y)p(x + y, z)(x + y + z) > p(x, y + z)(x + y + z). Thus δ2x,y = sup{0 < δ ≤ 1 :
p(x, y)p((1−δ)(x+y), z)((1−δ)(x+y)+z) > p(x, y+z)(x+y+z)} is well defined and for all
δ ∈ (0, δ2x,y), p(x, y)p((1− δ)(x+y), z)(1− δ)((1− δ)(x+y)+ z) > p(x, y+z)(1− δ)(x+y+z).
Let δ1(r) = minx,y∈Z(r) δ1x,y, δ
2(r) = minx,y∈Z(r) δ2x,y, and δ(r) = min(δ1(r), δ2(r)). Since Z(r)
is finite and non-empty so δ(r) is well defined and positive. For any δ ∈ (0, δ(r)) and amounts
of resources from Z(r), expected payoff from attacking a poorer side is higher then current
52
resource holding and expected payoff from attacking two opponents in a sequence is higher
than payoff from letting them fight and attacking them afterwards. Hence the argument in
proof of Theorem 1 works. This proves point 1.
Point 2: Let Zi = (1 − δ)ix0 +∑i
j=1(1 − δ)i−j+1xj. The expected payoff to a ruler with x0
resources from a sequence of conflicts with m rulers with resources x1, . . . , xm is given by
Πseq(x0, x1, . . . , xm) = Zm
m∏i=1
p (Zi−1, xi) = x0
m∏i=1
p (Zi−1, xi)ZiZi−1
.
Recall that if δ ≥ 1/2 then, for any x, y ∈ R++, (x+y)(1−δ)p(x, y) < x. Hence p(Zi−1, xi)Zi/Zi−1 =
p(Zi−1, xi)(1−δ)(Zi−1+xi)/Zi−1 < 1, for all i ∈ {1, . . . ,m} and, consequently, Πseq(x0, x1, . . . , xm) <
x0. Hence no non-empty sequence of fights is profitable if δ ≥ 1/2 and so there is peace in
equilibrium on any network and for any resource endowment r ∈ RV++. This completes the
proof of point 2.
Point 3: To prove the point, we will show first that with sufficiently large value of δ any
sequence of fights containing a fight with sufficiently small ruler yields lower expected payoff
than some of its subsequences ending before the fight with the small ruler. This observation
allows us to support the idea of buffer states in network settings.
Take any vector of resources (x0, . . . , xm) ∈ Rm+ and fix some k ∈ {0, . . . ,m}. Assume
that xi > 0, for all i 6= k. We will compare the expected payoffs from the sequence of fights
involving the sequence of resources x0, . . . , xm with sequences of fight involving the sequence
of resources x0, . . . , xl, for l ∈ {0, . . . , k − 1}.The inequality Πseq(x0, . . . , xm) < Πseq(x0, . . . , xl) is equivalent to
Zm
m∏i=l+1
p (Zi−1, xi) < Zl. (28)
We will show first, that the inequality (28) is satisfied if δ is sufficiently close to 1/2. Notice
that LHS = Πseq(Zl−1, xl, . . . , xm) and, by what we observed earlier, this is less than Zl−1,
if δ ≥ 1/2. Since the expected payoff is continuous in δ so, for any l ∈ {0, . . . , k − 1} and
xk ∈ R+, there exists δ(l, xk) ∈ [0, 1/2) such that (28) is satisfied for all δ ∈ [0, δ(l, xk)). Let
δ∗(xk) = minl∈{0,...,k−1} δ(l, xk). For any δ ∈ (δ∗(xk), 1/2) there exists l ∈ {0, . . . , k − 1} such
that the sequence of fights with resources x0, . . . , xl yields higher expected payoff than the
sequence of fights x0, . . . , xm.
Second, we show that the LHS is strictly increasing in xk on the interval (0, x) whith x
53
sufficiently close to 0. Notice that
∂LHS
∂xk= (1− δ)−l
(m∏
i=l+1
p (Zi−1, xi)
)((1− δ)m−k+1
− Zm
(f ′(xk)
f(xk)p(xk, Zk−1)−
m∑i=k+1
f ′(Zi−1)
f(Zi−1)p(xi, Zi−1)(1− δ)i−k
)).
Since f ′(0) = 0 so
f ′(xk)
f(xk)p(xk, Zk−1)
∣∣∣∣xk=0
=f ′(xk)
f(xk) + f(Zk−1)
∣∣∣∣xk=0
= 0, while
m∑i=k+1
f ′(Zi−1)
f(Zi−1)p(xi, Zi−1)(1− δ)i−k
∣∣∣∣∣xk=0
> 0.
Since f ′ is continuous around 0 so, for any δ ∈ [0, 1), there exists x(δ) > 0 such that ∂LHS∂xk
> 0
on (0, x(δ)) and, consequently, LHS is increasing in xk on (0, x(δ)). Let x∗ = infδ∈[0,1/2) x(δ):
for any δ ∈ [0, 1/2), LHS is increasing in xk on (0, x∗).
The two observations above allow us to conclude the following: any sequence of fights,
x1, . . . , xm with k ∈ {0, . . . ,m} yields lower expected payoff than one of its sequences, that
ends after l ≤ k − 1 fights if xk ∈ (0, x∗) and δ ∈ (δ∗(x∗), 1/2). Moreover, if δ∗ > 0, then
the sequence of length m yields weakly higher payoff than any of its subsequences of length
l ≤ k − 1 and yields exactly the same payoff as at least one of these subsequences. In this
case, increasing xk within the interval (0, x∗) will make the sequence of length m yield strictly
higher payoff than any of the subsequences of length l ≤ k − 1.
Now suppose that G has a node, v such that G− {v} is disconnected. By the conclusion
above, there exists a range (δ∗∗, 1/2) of losses in war and a range (0, x∗∗) of amounts of
resources such that in equilibrium no sequence of fights involves a fight with the ruler owning
v if resources at v, rv ∈ (0, x∗∗). It is important to note that the requirement on the resources
at v is necessary. With δ = 0, for any value of rv ∈ R++, there exists at least one strong ruler
whose full attacking sequence is better than any other sequence. By the analysis above, there
exists δ and r such that this sequence (or a subsequence of it) involves v, if rv > r and does
not involve v if rv < r.
This implies that buffer states prevent spread of conflict, if their resources are sufficiently
low and losses in war are sufficiently high.
54
Additional results
Hirshleifer’s constest success function
Another widely used contest success function, along the Tullock contest success function, is
the so called difference form proposed by Hirshleifer [1989]:
p(x, y) =exp(γx)
exp(γx) + exp(γy), (29)
where γ > 0. Thus f(x) = exp(γx) and it is easy to check that f(x)/x is increasing on interval
(0, 1/γ) and decreasing on (1/γ,+∞). Thus the function maintains the poor rewarding and,
consequently, the waiting properties on the interval (0, 1/γ) and maintains the rich rewarding
and, consequently, the no-waiting properties on the interval (1/γ,+∞). Hence if the minimal
resources in the network at the initial ownership configuration are greater than 1/γ, all the
results obtained for the rich rewarding case would hold for this contest success function as well
and if the total resources in the network are less than 1/γ, the results for the poor rewarding
case apply.
For the order of fights properties of Hirshleifer’s contest success function, notice that
h(s, t) =exp(γt) exp(γ(s+ t))
exp(γ(s+ t)− exp(γs)− exp(γt))=
1
exp(−γt)− exp(−2γt)− exp(−γ(s+ t)).
Taking the derivative with respect to t and comparing it to 0 we can see that h(s, t) is
decreasing in t when exp(−γt) < 1/2− exp(−γs)/2 and is increasing in t when the inequality
is reversed. The LHS of the inequality is decreasing in t while the RHS is increasing in s.
Moreover, the functions exp(−γx) and 1/2 − exp(−γx)/2 intersect at x = ln(3)/γ > 1/γ.
Thus on the interval (0, 1/γ) the contest success function maintains the poor rewarding and
the rich first properties and on interval (ln(3)/γ,+∞) it maintains the rich rewarding and the
poor first property.24
Greater losses lead to more war
Consider a star network over 4 vertices, as presented in Figure 10. Assume γ = 0.5. Every
spoke is endowed with y resources and the centre is endowed with x resources. Let y = 1.0
24Notice that on the interval (1/γ, ln(3)/γ) the contest success function is rich rewarding, but does nothave the poor first property: for any s ∈ (1/γ, ln(3)/γ), h(s, t) is first decreasing and then increasing in t on(1/γ, ln(3)/γ).
55
Figure 10: Network where increasing cost of conflict leads to war.
and x ∈ (2.1, 2.9).
Suppose that cost of conflict δ = 0. The expected payoff to a spoke ruler with 1.0 resources
from executing an attacking sequence of length m ≤ 2 when the centre ruler has z ∈ (3.1, 3.9)
resources is ϕ(z, 1.0,m) ≥ ϕ(z, 1.0, 2) ∈ (1.23, 1.37) (recall function ϕ as defined in proof of
Proposition 4). Hence after any spoke ruler attacks the centre at the initial ownership config-
uration, there will be fight till hegemony in any equilibrium. Payoff to the spoke ruler from
executing an attacking sequence of length 3 at the initial state is ϕ(x, 1.0, 3) ∈ (0.889, 0.999).
Thus it is not profitable for a spoke ruler to attack the centre at the initial state. Since γ < 1
and x < y so it is not profitable for the centre ruler to attack as well. Hence there is an
equilibrium with peace at the initial state.
Suppose now that cost of conflict δ = 0.2. The expected payoff to a spoke ruler with
y resources from executing an attacking sequence of length m when the centre ruler has x
resources is
ψ(x, y,m | δ) =(x(1− δ)m +
m∑j=1
(1− δ)jy
)p(y, x)
m−1∏i=1
p
((1− δ)ix+
i∑j=1
(1− δ)jy), y
). (30)
Consider the ownership configuration resulting from two attacks by spoke on a centre: there
are two active rulers, one with 1.0 resources and another one with z = 0.8(0.8(1.0+x)+1.0) ∈(2.784, 3.296) resources. Expected payoff to the poorer ruler from attacking the richer ruler is
ψ(z, 1.0, 1 | 2.0) ∈ (1.13, 1.23). Hence the poorer ruler finds it profitable to attack the richer
one. Consider now the ownership configuration resulting from one attack by a spoke ruler on
the centre. There are two spokes, each endowed with 1.0 resources and the centre endowed
with z = 0.8(1.0 + x) ∈ (2.48, 3.12) resources. Any attacker anticipates two fights after an
56
attack. Expected payoff to a spoke from two fights is ψ(z, 1.0, 2 | 2.0) ∈ (0.73, 0.81). Thus
a spoke ruler does not want to attack and (with γ < 1 and z > y) the centre ruler does
not want to attack as well. Lastly, consider the initial ownership configuration. Payoff to a
spoke from attacking the centre is ψ(x, 1.0, 1 | 2.0) ∈ (1.01, 1.16). This leads to an ownership
configuration with peace. Hence every spoke finds it profitable to attack the centre and so
there is no peace at the initial ownership configuration. The increase in the cost of conflict
prevents conflict escalation at states with less than four rulers. This in turn raises incentives
for rulers to attack in the initial state.
57